% version Nov 30
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{filecontents*}{llave-fig1.eps}
%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Adobe Illustrator(r) 6.0
%%For: (combs) (Dept of Math, Univ of Texas )
%%Title: (llave-fig1.eps)
%%CreationDate: (5/6/99) (8:19 AM)
%%BoundingBox: 144 465 449 615
%%HiResBoundingBox: 144 465 448.7293 615
%%DocumentProcessColors: Black
%%DocumentFonts: Helvetica
%%+ Symbol
%%+ Times-Italic
%%DocumentSuppliedResources: procset Adobe_level2_AI5 1.0 0
%%+ procset Adobe_typography_AI5 1.0 0
%%+ procset Adobe_Illustrator_AI6_vars Adobe_Illustrator_AI6
%%+ procset Adobe_Illustrator_AI5 1.0 0
%AI5_FileFormat 2.0
%AI3_ColorUsage: Black&White
%%AI6_ColorSeparationSet: 1 1 (AI6 Default Color Separation Set)
%%+ Options: 1 16 0 1 0 1 1 1 0 1 1 1 1 18 0 0 0 0 0 0 0 0 -1 -1
%%+ PPD: 1 21 0 0 60 45 2 2 1 0 0 1 0 0 0 0 0 0 0 0 0 0 ()
%AI3_TemplateBox: 306 396 306 396
%AI3_TileBox: 30 31 582 761
%AI3_DocumentPreview: Macintosh_ColorPic
%AI5_ArtSize: 612 792
%AI5_RulerUnits: 0
%AI5_ArtFlags: 1 0 0 1 0 0 1 1 0
%AI5_TargetResolution: 800
%AI5_NumLayers: 1
%AI5_OpenToView: 2 684 1.5 794 557 58 1 1 3 40
%AI5_OpenViewLayers: 7
%%EndComments
%%BeginProlog
%%BeginResource: procset Adobe_level2_AI5 1.2 0
%%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation)
%%Version: 1.2
%%CreationDate: (04/10/93) ()
%%Copyright: ((C) 1987-1993 Adobe Systems Incorporated All Rights Reserved)
userdict /Adobe_level2_AI5 23 dict dup begin
put
/packedarray where not
{
userdict begin
/packedarray
{
array astore readonly
} bind def
/setpacking /pop load def
/currentpacking false def
end
0
} if
pop
userdict /defaultpacking currentpacking put true setpacking
/initialize
{
Adobe_level2_AI5 begin
} bind def
/terminate
{
currentdict Adobe_level2_AI5 eq
{
end
} if
} bind def
mark
/setcustomcolor where not
{
/findcmykcustomcolor
{
5 packedarray
} bind def
/setcustomcolor
{
exch aload pop pop
4
{
4 index mul 4 1 roll
} repeat
5 -1 roll pop
setcmykcolor
}
def
} if
/gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def
userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put
userdict /level2?
systemdict /languagelevel known dup
{
pop systemdict /languagelevel get 2 ge
} if
put
/level2ScreenFreq
{
begin
60
HalftoneType 1 eq
{
pop Frequency
} if
HalftoneType 2 eq
{
pop GrayFrequency
} if
HalftoneType 5 eq
{
pop Default level2ScreenFreq
} if
end
} bind def
userdict /currentScreenFreq
level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put
level2? not
{
/setcmykcolor where not
{
/setcmykcolor
{
exch .11 mul add exch .59 mul add exch .3 mul add
1 exch sub setgray
} def
} if
/currentcmykcolor where not
{
/currentcmykcolor
{
0 0 0 1 currentgray sub
} def
} if
/setoverprint where not
{
/setoverprint /pop load def
} if
/selectfont where not
{
/selectfont
{
exch findfont exch
dup type /arraytype eq
{
makefont
}
{
scalefont
} ifelse
setfont
} bind def
} if
/cshow where not
{
/cshow
{
[
0 0 5 -1 roll aload pop
] cvx bind forall
} bind def
} if
} if
cleartomark
/anyColor?
{
add add add 0 ne
} bind def
/testColor
{
gsave
setcmykcolor currentcmykcolor
grestore
} bind def
/testCMYKColorThrough
{
testColor anyColor?
} bind def
userdict /composite?
level2?
{
gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore
add add add 4 eq
}
{
1 0 0 0 testCMYKColorThrough
0 1 0 0 testCMYKColorThrough
0 0 1 0 testCMYKColorThrough
0 0 0 1 testCMYKColorThrough
and and and
} ifelse
put
composite? not
{
userdict begin
gsave
/cyan? 1 0 0 0 testCMYKColorThrough def
/magenta? 0 1 0 0 testCMYKColorThrough def
/yellow? 0 0 1 0 testCMYKColorThrough def
/black? 0 0 0 1 testCMYKColorThrough def
grestore
/isCMYKSep? cyan? magenta? yellow? black? or or or def
/customColor? isCMYKSep? not def
end
} if
end defaultpacking setpacking
%%EndResource
%%BeginResource: procset Adobe_typography_AI5 1.0 1
%%Title: (Typography Operators)
%%Version: 1.0
%%CreationDate:(03/26/93) ()
%%Copyright: ((C) 1987-1993 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_typography_AI5 54 dict dup begin
put
/initialize
{
begin
begin
Adobe_typography_AI5 begin
Adobe_typography_AI5
{
dup xcheck
{
bind
} if
pop pop
} forall
end
end
end
Adobe_typography_AI5 begin
} def
/terminate
{
currentdict Adobe_typography_AI5 eq
{
end
} if
} def
/modifyEncoding
{
/_tempEncode exch ddef
/_pntr 0 ddef
{
counttomark -1 roll
dup type dup /marktype eq
{
pop pop exit
}
{
/nametype eq
{
_tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll
put
}
{
/_pntr exch ddef
} ifelse
} ifelse
} loop
_tempEncode
} def
/TE
{
StandardEncoding 256 array copy modifyEncoding
/_nativeEncoding exch def
} def
%
/TZ
{
dup type /arraytype eq
{
/_wv exch def
}
{
/_wv 0 def
} ifelse
/_useNativeEncoding exch def
pop pop
findfont _wv type /arraytype eq
{
_wv makeblendedfont
} if
dup length 2 add dict
begin
mark exch
{
1 index /FID ne
{
def
} if
cleartomark mark
} forall
pop
/FontName exch def
counttomark 0 eq
{
1 _useNativeEncoding eq
{
/Encoding _nativeEncoding def
} if
cleartomark
}
{
/Encoding load 256 array copy
modifyEncoding /Encoding exch def
} ifelse
FontName currentdict
end
definefont pop
} def
/tr
{
_ax _ay 3 2 roll
} def
/trj
{
_cx _cy _sp _ax _ay 6 5 roll
} def
/a0
{
/Tx
{
dup
currentpoint 3 2 roll
tr _psf
newpath moveto
tr _ctm _pss
} ddef
/Tj
{
dup
currentpoint 3 2 roll
trj _pjsf
newpath moveto
trj _ctm _pjss
} ddef
} def
/a1
{
/Tx
{
dup currentpoint 4 2 roll gsave
dup currentpoint 3 2 roll
tr _psf
newpath moveto
tr _ctm _pss
grestore 3 1 roll moveto tr sp
} ddef
/Tj
{
dup currentpoint 4 2 roll gsave
dup currentpoint 3 2 roll
trj _pjsf
newpath moveto
trj _ctm _pjss
grestore 3 1 roll moveto tr jsp
} ddef
} def
/e0
{
/Tx
{
tr _psf
} ddef
/Tj
{
trj _pjsf
} ddef
} def
/e1
{
/Tx
{
dup currentpoint 4 2 roll gsave
tr _psf
grestore 3 1 roll moveto tr sp
} ddef
/Tj
{
dup currentpoint 4 2 roll gsave
trj _pjsf
grestore 3 1 roll moveto tr jsp
} ddef
} def
/i0
{
/Tx
{
tr sp
} ddef
/Tj
{
trj jsp
} ddef
} def
/i1
{
W N
} def
/o0
{
/Tx
{
tr sw rmoveto
} ddef
/Tj
{
trj swj rmoveto
} ddef
} def
/r0
{
/Tx
{
tr _ctm _pss
} ddef
/Tj
{
trj _ctm _pjss
} ddef
} def
/r1
{
/Tx
{
dup currentpoint 4 2 roll currentpoint gsave newpath moveto
tr _ctm _pss
grestore 3 1 roll moveto tr sp
} ddef
/Tj
{
dup currentpoint 4 2 roll currentpoint gsave newpath moveto
trj _ctm _pjss
grestore 3 1 roll moveto tr jsp
} ddef
} def
/To
{
pop _ctm currentmatrix pop
} def
/TO
{
iTe _ctm setmatrix newpath
} def
/Tp
{
pop _tm astore pop _ctm setmatrix
_tDict begin
/W
{
} def
/h
{
} def
} def
/TP
{
end
iTm 0 0 moveto
} def
/Tr
{
_render 3 le
{
currentpoint newpath moveto
} if
dup 8 eq
{
pop 0
}
{
dup 9 eq
{
pop 1
} if
} ifelse
dup /_render exch ddef
_renderStart exch get load exec
} def
/iTm
{
_ctm setmatrix _tm concat 0 _rise translate _hs 1 scale
} def
/Tm
{
_tm astore pop iTm 0 0 moveto
} def
/Td
{
_mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto
} def
/iTe
{
_render -1 eq
{
}
{
_renderEnd _render get dup null ne
{
load exec
}
{
pop
} ifelse
} ifelse
/_render -1 ddef
} def
/Ta
{
pop
} def
/Tf
{
dup 1000 div /_fScl exch ddef
%
selectfont
} def
/Tl
{
pop
0 exch _leading astore pop
} def
/Tt
{
pop
} def
/TW
{
3 npop
} def
/Tw
{
/_cx exch ddef
} def
/TC
{
3 npop
} def
/Tc
{
/_ax exch ddef
} def
/Ts
{
/_rise exch ddef
currentpoint
iTm
moveto
} def
/Ti
{
3 npop
} def
/Tz
{
100 div /_hs exch ddef
iTm
} def
/TA
{
pop
} def
/Tq
{
pop
} def
/Th
{
pop pop pop pop pop
} def
/TX
{
pop
} def
/Tk
{
exch pop _fScl mul neg 0 rmoveto
} def
/TK
{
2 npop
} def
/T*
{
_leading aload pop neg Td
} def
/T*-
{
_leading aload pop Td
} def
/T-
{
_ax neg 0 rmoveto
_hyphen Tx
} def
/T+
{
} def
/TR
{
_ctm currentmatrix pop
_tm astore pop
iTm 0 0 moveto
} def
/TS
{
currentfont 3 1 roll
/_Symbol_ _fScl 1000 mul selectfont
0 eq
{
Tx
}
{
Tj
} ifelse
setfont
} def
/Xb
{
pop pop
} def
/Tb /Xb load def
/Xe
{
pop pop pop pop
} def
/Te /Xe load def
/XB
{
} def
/TB /XB load def
currentdict readonly pop
end
setpacking
%%EndResource
%%BeginProcSet: Adobe_ColorImage_AI6 1.0 0
userdict /Adobe_ColorImage_AI6 known not
{
userdict /Adobe_ColorImage_AI6 17 dict put
} if
userdict /Adobe_ColorImage_AI6 get begin
/initialize
{
Adobe_ColorImage_AI6 begin
Adobe_ColorImage_AI6
{
dup type /arraytype eq
{
dup xcheck
{
bind
} if
} if
pop pop
} forall
} def
/terminate { end } def
currentdict /Adobe_ColorImage_AI6_Vars known not
{
/Adobe_ColorImage_AI6_Vars 14 dict def
} if
Adobe_ColorImage_AI6_Vars begin
/channelcount 0 def
/sourcecount 0 def
/sourcearray 4 array def
/plateindex -1 def
/XIMask 0 def
/XIBinary 0 def
/XIChannelCount 0 def
/XIBitsPerPixel 0 def
/XIImageHeight 0 def
/XIImageWidth 0 def
/XIImageMatrix null def
/XIBuffer null def
/XIDataProc null def
end
/WalkRGBString null def
/WalkCMYKString null def
/StuffRGBIntoGrayString null def
/RGBToGrayImageProc null def
/StuffCMYKIntoGrayString null def
/CMYKToGrayImageProc null def
/ColorImageCompositeEmulator null def
/SeparateCMYKImageProc null def
/FourEqual null def
/TestPlateIndex null def
currentdict /_colorimage known not
{
/colorimage where
{
/colorimage get /_colorimage exch def
}
{
/_colorimage null def
} ifelse
} if
/_currenttransfer systemdict /currenttransfer get def
/colorimage null def
/XI null def
/WalkRGBString
{
0 3 index
dup length 1 sub 0 3 3 -1 roll
{
3 getinterval { } forall
5 index exec
3 index
} for
5 { pop } repeat
} def
/WalkCMYKString
{
0 3 index
dup length 1 sub 0 4 3 -1 roll
{
4 getinterval { } forall
6 index exec
3 index
} for
5 { pop } repeat
} def
/StuffRGBIntoGrayString
{
.11 mul exch
.59 mul add exch
.3 mul add
cvi 3 copy put
pop 1 add
} def
/RGBToGrayImageProc
{
Adobe_ColorImage_AI6_Vars begin
sourcearray 0 get exec
dup length 3 idiv string
dup 3 1 roll
/StuffRGBIntoGrayString load exch
WalkRGBString
end
} def
/StuffCMYKIntoGrayString
{
exch .11 mul add
exch .59 mul add
exch .3 mul add
dup 255 gt { pop 255 } if
255 exch sub cvi 3 copy put
pop 1 add
} def
/CMYKToGrayImageProc
{
Adobe_ColorImage_AI6_Vars begin
sourcearray 0 get exec
dup length 4 idiv string
dup 3 1 roll
/StuffCMYKIntoGrayString load exch
WalkCMYKString
end
} def
/ColorImageCompositeEmulator
{
pop true eq
{
Adobe_ColorImage_AI6_Vars /sourcecount get 5 add { pop } repeat
}
{
Adobe_ColorImage_AI6_Vars /channelcount get 1 ne
{
Adobe_ColorImage_AI6_Vars begin
sourcearray 0 3 -1 roll put
channelcount 3 eq
{
/RGBToGrayImageProc
}
{
/CMYKToGrayImageProc
} ifelse
load
end
} if
image
} ifelse
} def
/SeparateCMYKImageProc
{
Adobe_ColorImage_AI6_Vars begin
sourcecount 0 ne
{
sourcearray plateindex get exec
}
{
sourcearray 0 get exec
dup length 4 idiv string
0 2 index
plateindex 4 2 index length 1 sub
{
get 255 exch sub
3 copy put pop 1 add
2 index
} for
pop pop exch pop
} ifelse
end
} def
/FourEqual
{
4 index ne
{
pop pop pop false
}
{
4 index ne
{
pop pop false
}
{
4 index ne
{
pop false
}
{
4 index eq
} ifelse
} ifelse
} ifelse
} def
/TestPlateIndex
{
Adobe_ColorImage_AI6_Vars begin
/plateindex -1 def
/setcmykcolor where
{
pop
gsave
1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub
grestore
1 0 0 0 FourEqual
{
/plateindex 0 def
}
{
0 1 0 0 FourEqual
{
/plateindex 1 def
}
{
0 0 1 0 FourEqual
{
/plateindex 2 def
}
{
0 0 0 1 FourEqual
{
/plateindex 3 def
}
{
0 0 0 0 FourEqual
{
/plateindex 5 def
} if
} ifelse
} ifelse
} ifelse
} ifelse
pop pop pop pop
} if
plateindex
end
} def
/colorimage
{
Adobe_ColorImage_AI6_Vars begin
/channelcount 1 index def
/sourcecount 2 index 1 eq { channelcount 1 sub } { 0 } ifelse def
4 sourcecount add index dup
8 eq exch 1 eq or not
end
{
/_colorimage load null ne
{
_colorimage
}
{
Adobe_ColorImage_AI6_Vars /sourcecount get
7 add { pop } repeat
} ifelse
}
{
dup 3 eq
TestPlateIndex
dup -1 eq exch 5 eq or or
{
/_colorimage load null eq
{
ColorImageCompositeEmulator
}
{
dup 1 eq
{
pop pop image
}
{
Adobe_ColorImage_AI6_Vars /plateindex get 5 eq
{
gsave
0 _currenttransfer exec
1 _currenttransfer exec
eq
{ 0 _currenttransfer exec 0.5 lt }
{ 0 _currenttransfer exec 1 _currenttransfer exec gt } ifelse
{ { pop 0 } } { { pop 1 } } ifelse
systemdict /settransfer get exec
} if
_colorimage
Adobe_ColorImage_AI6_Vars /plateindex get 5 eq
{
grestore
} if
} ifelse
} ifelse
}
{
dup 1 eq
{
pop pop
image
}
{
pop pop
Adobe_ColorImage_AI6_Vars begin
sourcecount -1 0
{
exch sourcearray 3 1 roll put
} for
/SeparateCMYKImageProc load
end
systemdict /image get exec
} ifelse
} ifelse
} ifelse
} def
/XI
{
Adobe_ColorImage_AI6_Vars begin
gsave
/XIMask exch 0 ne def
/XIBinary exch 0 ne def
pop
pop
/XIChannelCount exch def
/XIBitsPerPixel exch def
/XIImageHeight exch def
/XIImageWidth exch def
pop pop pop pop
/XIImageMatrix exch def
XIBitsPerPixel 1 eq
{
XIImageWidth 8 div ceiling cvi
}
{
XIImageWidth XIChannelCount mul
} ifelse
/XIBuffer exch string def
XIBinary
{
/XIDataProc { currentfile XIBuffer readstring pop } def
currentfile 128 string readline pop pop
}
{
/XIDataProc { currentfile XIBuffer readhexstring pop } def
} ifelse
0 0 moveto
XIImageMatrix concat
XIImageWidth XIImageHeight scale
XIMask
{
XIImageWidth XIImageHeight
false
[ XIImageWidth 0 0 XIImageHeight neg 0 0 ]
/XIDataProc load
/_lp /null ddef
_fc
/_lp /imagemask ddef
imagemask
}
{
XIImageWidth XIImageHeight
XIBitsPerPixel
[ XIImageWidth 0 0 XIImageHeight neg 0 0 ]
/XIDataProc load
XIChannelCount 1 eq
{
gsave
0 setgray
image
grestore
}
{
false
XIChannelCount
colorimage
} ifelse
} ifelse
grestore
end
} def
end
%%EndProcSet
%%BeginResource: procset Adobe_Illustrator_AI5 1.1 0
%%Title: (Adobe Illustrator (R) Version 5.0 Full Prolog)
%%Version: 1.1
%%CreationDate: (3/7/1994) ()
%%Copyright: ((C) 1987-1994 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_Illustrator_AI5_vars 81 dict dup begin
put
/_eo false def
/_lp /none def
/_pf
{
} def
/_ps
{
} def
/_psf
{
} def
/_pss
{
} def
/_pjsf
{
} def
/_pjss
{
} def
/_pola 0 def
/_doClip 0 def
/cf currentflat def
/_tm matrix def
/_renderStart
[
/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0
] def
/_renderEnd
[
null null null null /i1 /i1 /i1 /i1
] def
/_render -1 def
/_rise 0 def
/_ax 0 def
/_ay 0 def
/_cx 0 def
/_cy 0 def
/_leading
[
0 0
] def
/_ctm matrix def
/_mtx matrix def
/_sp 16#020 def
/_hyphen (-) def
/_fScl 0 def
/_cnt 0 def
/_hs 1 def
/_nativeEncoding 0 def
/_useNativeEncoding 0 def
/_tempEncode 0 def
/_pntr 0 def
/_tDict 2 dict def
/_wv 0 def
/Tx
{
} def
/Tj
{
} def
/CRender
{
} def
/_AI3_savepage
{
} def
/_gf null def
/_cf 4 array def
/_if null def
/_of false def
/_fc
{
} def
/_gs null def
/_cs 4 array def
/_is null def
/_os false def
/_sc
{
} def
/_pd 1 dict def
/_ed 15 dict def
/_pm matrix def
/_fm null def
/_fd null def
/_fdd null def
/_sm null def
/_sd null def
/_sdd null def
/_i null def
/discardSave null def
/buffer 256 string def
/beginString null def
/endString null def
/endStringLength null def
/layerCnt 1 def
/layerCount 1 def
/perCent (%) 0 get def
/perCentSeen? false def
/newBuff null def
/newBuffButFirst null def
/newBuffLast null def
/clipForward? false def
end
userdict /Adobe_Illustrator_AI5 known not {
userdict /Adobe_Illustrator_AI5 91 dict put
} if
userdict /Adobe_Illustrator_AI5 get begin
/initialize
{
Adobe_Illustrator_AI5 dup begin
Adobe_Illustrator_AI5_vars begin
discardDict
{
bind pop pop
} forall
dup /nc get begin
{
dup xcheck 1 index type /operatortype ne and
{
bind
} if
pop pop
} forall
end
newpath
} def
/terminate
{
end
end
} def
/_
null def
/ddef
{
Adobe_Illustrator_AI5_vars 3 1 roll put
} def
/xput
{
dup load dup length exch maxlength eq
{
dup dup load dup
length 2 mul dict copy def
} if
load begin
def
end
} def
/npop
{
{
pop
} repeat
} def
/sw
{
dup length exch stringwidth
exch 5 -1 roll 3 index mul add
4 1 roll 3 1 roll mul add
} def
/swj
{
dup 4 1 roll
dup length exch stringwidth
exch 5 -1 roll 3 index mul add
4 1 roll 3 1 roll mul add
6 2 roll /_cnt 0 ddef
{
1 index eq
{
/_cnt _cnt 1 add ddef
} if
} forall
pop
exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop
} def
/ss
{
4 1 roll
{
2 npop
(0) exch 2 copy 0 exch put pop
gsave
false charpath currentpoint
4 index setmatrix
stroke
grestore
moveto
2 copy rmoveto
} exch cshow
3 npop
} def
/jss
{
4 1 roll
{
2 npop
(0) exch 2 copy 0 exch put
gsave
_sp eq
{
exch 6 index 6 index 6 index 5 -1 roll widthshow
currentpoint
}
{
false charpath currentpoint
4 index setmatrix stroke
} ifelse
grestore
moveto
2 copy rmoveto
} exch cshow
6 npop
} def
/sp
{
{
2 npop (0) exch
2 copy 0 exch put pop
false charpath
2 copy rmoveto
} exch cshow
2 npop
} def
/jsp
{
{
2 npop
(0) exch 2 copy 0 exch put
_sp eq
{
exch 5 index 5 index 5 index 5 -1 roll widthshow
}
{
false charpath
} ifelse
2 copy rmoveto
} exch cshow
5 npop
} def
/pl
{
transform
0.25 sub round 0.25 add exch
0.25 sub round 0.25 add exch
itransform
} def
/setstrokeadjust where
{
pop true setstrokeadjust
/c
{
curveto
} def
/C
/c load def
/v
{
currentpoint 6 2 roll curveto
} def
/V
/v load def
/y
{
2 copy curveto
} def
/Y
/y load def
/l
{
lineto
} def
/L
/l load def
/m
{
moveto
} def
}
{
/c
{
pl curveto
} def
/C
/c load def
/v
{
currentpoint 6 2 roll pl curveto
} def
/V
/v load def
/y
{
pl 2 copy curveto
} def
/Y
/y load def
/l
{
pl lineto
} def
/L
/l load def
/m
{
pl moveto
} def
} ifelse
/d
{
setdash
} def
/cf
{
} def
/i
{
dup 0 eq
{
pop cf
} if
setflat
} def
/j
{
setlinejoin
} def
/J
{
setlinecap
} def
/M
{
setmiterlimit
} def
/w
{
setlinewidth
} def
/XR
{
0 ne
/_eo exch ddef
} def
/H
{
} def
/h
{
closepath
} def
/N
{
_pola 0 eq
{
_doClip 1 eq
{
_eo {eoclip} {clip} ifelse /_doClip 0 ddef
} if
newpath
}
{
/CRender
{
N
} ddef
} ifelse
} def
/n
{
N
} def
/F
{
_pola 0 eq
{
_doClip 1 eq
{
gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc
/_doClip 0 ddef
}
{
_pf
} ifelse
}
{
/CRender
{
F
} ddef
} ifelse
} def
/f
{
closepath
F
} def
/S
{
_pola 0 eq
{
_doClip 1 eq
{
gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc
/_doClip 0 ddef
}
{
_ps
} ifelse
}
{
/CRender
{
S
} ddef
} ifelse
} def
/s
{
closepath
S
} def
/B
{
_pola 0 eq
{
_doClip 1 eq
gsave F grestore
{
gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc
/_doClip 0 ddef
}
{
S
} ifelse
}
{
/CRender
{
B
} ddef
} ifelse
} def
/b
{
closepath
B
} def
/W
{
/_doClip 1 ddef
} def
/*
{
count 0 ne
{
dup type /stringtype eq
{
pop
} if
} if
newpath
} def
/u
{
} def
/U
{
} def
/q
{
_pola 0 eq
{
gsave
} if
} def
/Q
{
_pola 0 eq
{
grestore
} if
} def
/*u
{
_pola 1 add /_pola exch ddef
} def
/*U
{
_pola 1 sub /_pola exch ddef
_pola 0 eq
{
CRender
} if
} def
/D
{
pop
} def
/*w
{
} def
/*W
{
} def
/`
{
/_i save ddef
clipForward?
{
nulldevice
} if
6 1 roll 4 npop
concat pop
userdict begin
/showpage
{
} def
0 setgray
0 setlinecap
1 setlinewidth
0 setlinejoin
10 setmiterlimit
[] 0 setdash
/setstrokeadjust where {pop false setstrokeadjust} if
newpath
0 setgray
false setoverprint
} def
/~
{
end
_i restore
} def
/O
{
0 ne
/_of exch ddef
/_lp /none ddef
} def
/R
{
0 ne
/_os exch ddef
/_lp /none ddef
} def
/g
{
/_gf exch ddef
/_fc
{
_lp /fill ne
{
_of setoverprint
_gf setgray
/_lp /fill ddef
} if
} ddef
/_pf
{
_fc
_eo {eofill} {fill} ifelse
} ddef
/_psf
{
_fc
ashow
} ddef
/_pjsf
{
_fc
awidthshow
} ddef
/_lp /none ddef
} def
/G
{
/_gs exch ddef
/_sc
{
_lp /stroke ne
{
_os setoverprint
_gs setgray
/_lp /stroke ddef
} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
/k
{
_cf astore pop
/_fc
{
_lp /fill ne
{
_of setoverprint
_cf aload pop setcmykcolor
/_lp /fill ddef
} if
} ddef
/_pf
{
_fc
_eo {eofill} {fill} ifelse
} ddef
/_psf
{
_fc
ashow
} ddef
/_pjsf
{
_fc
awidthshow
} ddef
/_lp /none ddef
} def
/K
{
_cs astore pop
/_sc
{
_lp /stroke ne
{
_os setoverprint
_cs aload pop setcmykcolor
/_lp /stroke ddef
} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
/x
{
/_gf exch ddef
findcmykcustomcolor
/_if exch ddef
/_fc
{
_lp /fill ne
{
_of setoverprint
_if _gf 1 exch sub setcustomcolor
/_lp /fill ddef
} if
} ddef
/_pf
{
_fc
_eo {eofill} {fill} ifelse
} ddef
/_psf
{
_fc
ashow
} ddef
/_pjsf
{
_fc
awidthshow
} ddef
/_lp /none ddef
} def
/X
{
/_gs exch ddef
findcmykcustomcolor
/_is exch ddef
/_sc
{
_lp /stroke ne
{
_os setoverprint
_is _gs 1 exch sub setcustomcolor
/_lp /stroke ddef
} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
/A
{
pop
} def
/annotatepage
{
userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse
} def
/XT {
pop pop
} def
/discard
{
save /discardSave exch store
discardDict begin
/endString exch store
gt38?
{
2 add
} if
load
stopped
pop
end
discardSave restore
} bind def
userdict /discardDict 7 dict dup begin
put
/pre38Initialize
{
/endStringLength endString length store
/newBuff buffer 0 endStringLength getinterval store
/newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store
/newBuffLast newBuff endStringLength 1 sub 1 getinterval store
} def
/shiftBuffer
{
newBuff 0 newBuffButFirst putinterval
newBuffLast 0
currentfile read not
{
stop
} if
put
} def
0
{
pre38Initialize
mark
currentfile newBuff readstring exch pop
{
{
newBuff endString eq
{
cleartomark stop
} if
shiftBuffer
} loop
}
{
stop
} ifelse
} def
1
{
pre38Initialize
/beginString exch store
mark
currentfile newBuff readstring exch pop
{
{
newBuff beginString eq
{
/layerCount dup load 1 add store
}
{
newBuff endString eq
{
/layerCount dup load 1 sub store
layerCount 0 eq
{
cleartomark stop
} if
} if
} ifelse
shiftBuffer
} loop
} if
} def
2
{
mark
{
currentfile buffer readline not
{
stop
} if
endString eq
{
cleartomark stop
} if
} loop
} def
3
{
/beginString exch store
/layerCnt 1 store
mark
{
currentfile buffer readline not
{
stop
} if
dup beginString eq
{
pop /layerCnt dup load 1 add store
}
{
endString eq
{
layerCnt 1 eq
{
cleartomark stop
}
{
/layerCnt dup load 1 sub store
} ifelse
} if
} ifelse
} loop
} def
end
userdict /clipRenderOff 15 dict dup begin
put
{
/n /N /s /S /f /F /b /B
}
{
{
_doClip 1 eq
{
/_doClip 0 ddef _eo {eoclip} {clip} ifelse
} if
newpath
} def
} forall
/Tr /pop load def
/Bb {} def
/BB /pop load def
/Bg {12 npop} def
/Bm {6 npop} def
/Bc /Bm load def
/Bh {4 npop} def
end
/Lb
{
4 npop
6 1 roll
pop
4 1 roll
pop pop pop
0 eq
{
0 eq
{
(%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard
}
{
/clipForward? true def
/Tx /pop load def
/Tj /pop load def
currentdict end clipRenderOff begin begin
} ifelse
}
{
0 eq
{
save /discardSave exch store
} if
} ifelse
} bind def
/LB
{
discardSave dup null ne
{
restore
}
{
pop
clipForward?
{
currentdict
end
end
begin
/clipForward? false ddef
} if
} ifelse
} bind def
/Pb
{
pop pop
0 (%AI5_EndPalette) discard
} bind def
/Np
{
0 (%AI5_End_NonPrinting--) discard
} bind def
/Ln /pop load def
/Ap
/pop load def
/Ar
{
72 exch div
0 dtransform dup mul exch dup mul add sqrt
dup 1 lt
{
pop 1
} if
setflat
} def
/Mb
{
q
} def
/Md
{
} def
/MB
{
Q
} def
/nc 3 dict def
nc begin
/setgray
{
pop
} bind def
/setcmykcolor
{
4 npop
} bind def
/setcustomcolor
{
2 npop
} bind def
currentdict readonly pop
end
end
setpacking
%%EndResource
%%EndProlog
%%BeginSetup
%%IncludeFont: Helvetica
%%IncludeFont: Symbol
%%IncludeFont: Times-Italic
Adobe_level2_AI5 /initialize get exec
Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec
Adobe_ColorImage_AI6 /initialize get exec
Adobe_Illustrator_AI5 /initialize get exec
[
39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis
/Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute
/egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde
/oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex
/udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls
/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash
/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef
/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash
/questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef
/guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe
/endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide
/.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright
/fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand
/Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex
/Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex
/Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla
/hungarumlaut/ogonek/caron
TE
%AI3_BeginEncoding: _Helvetica Helvetica
[/_Helvetica/Helvetica 0 0 1 TZ
%AI3_EndEncoding AdobeType
%AI3_BeginEncoding: _Symbol Symbol
[/_Symbol/Symbol 0 0 0 TZ
%AI3_EndEncoding TrueType
%AI3_BeginEncoding: _Times-Italic Times-Italic
[/_Times-Italic/Times-Italic 0 0 1 TZ
%AI3_EndEncoding TrueType
%AI5_Begin_NonPrinting
Np
8 Bn
%AI5_BeginGradient: (Black & White)
(Black & White) 0 2 Bd
[
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(Green & Blue) 0 2 Bd
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426 516 l
S
357.5312 528.6875 m
382.9687 542.0625 l
S
153 546 m
153 526.5 l
S
166.5 543 m
166.5 523.5 l
S
186 544.5 m
186 525 l
S
175.5 541.5 m
175.5 522 l
S
249 535.5 m
312 535.5 l
S
262.5 526.5 m
325.5 526.5 l
S
246 516 m
309 516 l
S
255 501 m
318 501 l
S
0 To
1 0 0 1 394.6667 520 0 Tp
TP
0 Tr
0 O
0 g
/_Helvetica 36 Tf
0 Ts
100 Tz
0 Tt
1 TA
%_ 0 XL
36 0 Xb
XB
0 0 5 TC
100 100 200 TW
0 0 0 Ti
0 Ta
0 0 2 2 3 Th
0 Tq
0 0 Tl
0 Tc
0 Tw
(+) Tx
(\r) TX
TO
0 To
1 0 0 1 285 495 0 Tp
TP
0 Tr
(-) Tx
(\r) TX
TO
0 To
1 0 0 1 358 576 0 Tp
TP
0 Tr
/_Times-Italic 18 Tf
(T) Tx
/_Helvetica 18 Tf
(\() Tx
/_Symbol 18 Tf
(g) Tx
/_Helvetica 18 Tf
(\)) Tx
(\r) TX
TO
0 To
1 0 0 1 441.3333 493.3333 0 Tp
TP
0 Tr
/_Symbol 18 Tf
(g) Tx
(\r) TX
TO
LB
%AI5_EndLayer--
%%PageTrailer
gsave annotatepage grestore showpage
%%Trailer
Adobe_Illustrator_AI5 /terminate get exec
Adobe_ColorImage_AI6 /terminate get exec
Adobe_typography_AI5 /terminate get exec
Adobe_level2_AI5 /terminate get exec
%%EOF
\end{filecontents*}
\begin{filecontents*}{llave-fig2.eps}
%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Adobe Illustrator(r) 6.0
%%For: (combs) (Dept of Math, Univ of Texas )
%%Title: (llave-fig2.eps)
%%CreationDate: (5/13/99) (3:52 PM)
%%BoundingBox: 18 120 553 717
%%HiResBoundingBox: 18 120.5485 552.2673 716.4407
%%DocumentProcessColors: Black
%%DocumentFonts: Symbol
%%+ Times-Italic
%%+ Times-Roman
%%DocumentSuppliedResources: procset Adobe_level2_AI5 1.0 0
%%+ procset Adobe_typography_AI5 1.0 0
%%+ procset Adobe_Illustrator_AI6_vars Adobe_Illustrator_AI6
%%+ procset Adobe_Illustrator_AI5 1.0 0
%AI5_FileFormat 2.0
%AI3_ColorUsage: Black&White
%%AI6_ColorSeparationSet: 1 1 (AI6 Default Color Separation Set)
%%+ Options: 1 16 0 1 0 1 1 1 0 1 1 1 1 18 0 0 0 0 0 0 0 0 -1 -1
%%+ PPD: 1 21 0 0 60 45 2 2 1 0 0 1 0 0 0 0 0 0 0 0 0 0 ()
%AI3_TemplateBox: 306 396 306 396
%AI3_TileBox: 30 31 582 761
%AI3_DocumentPreview: Macintosh_ColorPic
%AI5_ArtSize: 612 792
%AI5_RulerUnits: 0
%AI5_ArtFlags: 1 0 0 1 0 0 1 1 0
%AI5_TargetResolution: 800
%AI5_NumLayers: 1
%AI5_OpenToView: 66 620 1.5 790 553 58 1 1 3 40
%AI5_OpenViewLayers: 7
%%EndComments
%%BeginProlog
%%BeginResource: procset Adobe_level2_AI5 1.2 0
%%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation)
%%Version: 1.2
%%CreationDate: (04/10/93) ()
%%Copyright: ((C) 1987-1993 Adobe Systems Incorporated All Rights Reserved)
userdict /Adobe_level2_AI5 23 dict dup begin
put
/packedarray where not
{
userdict begin
/packedarray
{
array astore readonly
} bind def
/setpacking /pop load def
/currentpacking false def
end
0
} if
pop
userdict /defaultpacking currentpacking put true setpacking
/initialize
{
Adobe_level2_AI5 begin
} bind def
/terminate
{
currentdict Adobe_level2_AI5 eq
{
end
} if
} bind def
mark
/setcustomcolor where not
{
/findcmykcustomcolor
{
5 packedarray
} bind def
/setcustomcolor
{
exch aload pop pop
4
{
4 index mul 4 1 roll
} repeat
5 -1 roll pop
setcmykcolor
}
def
} if
/gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def
userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put
userdict /level2?
systemdict /languagelevel known dup
{
pop systemdict /languagelevel get 2 ge
} if
put
/level2ScreenFreq
{
begin
60
HalftoneType 1 eq
{
pop Frequency
} if
HalftoneType 2 eq
{
pop GrayFrequency
} if
HalftoneType 5 eq
{
pop Default level2ScreenFreq
} if
end
} bind def
userdict /currentScreenFreq
level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put
level2? not
{
/setcmykcolor where not
{
/setcmykcolor
{
exch .11 mul add exch .59 mul add exch .3 mul add
1 exch sub setgray
} def
} if
/currentcmykcolor where not
{
/currentcmykcolor
{
0 0 0 1 currentgray sub
} def
} if
/setoverprint where not
{
/setoverprint /pop load def
} if
/selectfont where not
{
/selectfont
{
exch findfont exch
dup type /arraytype eq
{
makefont
}
{
scalefont
} ifelse
setfont
} bind def
} if
/cshow where not
{
/cshow
{
[
0 0 5 -1 roll aload pop
] cvx bind forall
} bind def
} if
} if
cleartomark
/anyColor?
{
add add add 0 ne
} bind def
/testColor
{
gsave
setcmykcolor currentcmykcolor
grestore
} bind def
/testCMYKColorThrough
{
testColor anyColor?
} bind def
userdict /composite?
level2?
{
gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore
add add add 4 eq
}
{
1 0 0 0 testCMYKColorThrough
0 1 0 0 testCMYKColorThrough
0 0 1 0 testCMYKColorThrough
0 0 0 1 testCMYKColorThrough
and and and
} ifelse
put
composite? not
{
userdict begin
gsave
/cyan? 1 0 0 0 testCMYKColorThrough def
/magenta? 0 1 0 0 testCMYKColorThrough def
/yellow? 0 0 1 0 testCMYKColorThrough def
/black? 0 0 0 1 testCMYKColorThrough def
grestore
/isCMYKSep? cyan? magenta? yellow? black? or or or def
/customColor? isCMYKSep? not def
end
} if
end defaultpacking setpacking
%%EndResource
%%BeginResource: procset Adobe_typography_AI5 1.0 1
%%Title: (Typography Operators)
%%Version: 1.0
%%CreationDate:(03/26/93) ()
%%Copyright: ((C) 1987-1993 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_typography_AI5 54 dict dup begin
put
/initialize
{
begin
begin
Adobe_typography_AI5 begin
Adobe_typography_AI5
{
dup xcheck
{
bind
} if
pop pop
} forall
end
end
end
Adobe_typography_AI5 begin
} def
/terminate
{
currentdict Adobe_typography_AI5 eq
{
end
} if
} def
/modifyEncoding
{
/_tempEncode exch ddef
/_pntr 0 ddef
{
counttomark -1 roll
dup type dup /marktype eq
{
pop pop exit
}
{
/nametype eq
{
_tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll
put
}
{
/_pntr exch ddef
} ifelse
} ifelse
} loop
_tempEncode
} def
/TE
{
StandardEncoding 256 array copy modifyEncoding
/_nativeEncoding exch def
} def
%
/TZ
{
dup type /arraytype eq
{
/_wv exch def
}
{
/_wv 0 def
} ifelse
/_useNativeEncoding exch def
pop pop
findfont _wv type /arraytype eq
{
_wv makeblendedfont
} if
dup length 2 add dict
begin
mark exch
{
1 index /FID ne
{
def
} if
cleartomark mark
} forall
pop
/FontName exch def
counttomark 0 eq
{
1 _useNativeEncoding eq
{
/Encoding _nativeEncoding def
} if
cleartomark
}
{
/Encoding load 256 array copy
modifyEncoding /Encoding exch def
} ifelse
FontName currentdict
end
definefont pop
} def
/tr
{
_ax _ay 3 2 roll
} def
/trj
{
_cx _cy _sp _ax _ay 6 5 roll
} def
/a0
{
/Tx
{
dup
currentpoint 3 2 roll
tr _psf
newpath moveto
tr _ctm _pss
} ddef
/Tj
{
dup
currentpoint 3 2 roll
trj _pjsf
newpath moveto
trj _ctm _pjss
} ddef
} def
/a1
{
/Tx
{
dup currentpoint 4 2 roll gsave
dup currentpoint 3 2 roll
tr _psf
newpath moveto
tr _ctm _pss
grestore 3 1 roll moveto tr sp
} ddef
/Tj
{
dup currentpoint 4 2 roll gsave
dup currentpoint 3 2 roll
trj _pjsf
newpath moveto
trj _ctm _pjss
grestore 3 1 roll moveto tr jsp
} ddef
} def
/e0
{
/Tx
{
tr _psf
} ddef
/Tj
{
trj _pjsf
} ddef
} def
/e1
{
/Tx
{
dup currentpoint 4 2 roll gsave
tr _psf
grestore 3 1 roll moveto tr sp
} ddef
/Tj
{
dup currentpoint 4 2 roll gsave
trj _pjsf
grestore 3 1 roll moveto tr jsp
} ddef
} def
/i0
{
/Tx
{
tr sp
} ddef
/Tj
{
trj jsp
} ddef
} def
/i1
{
W N
} def
/o0
{
/Tx
{
tr sw rmoveto
} ddef
/Tj
{
trj swj rmoveto
} ddef
} def
/r0
{
/Tx
{
tr _ctm _pss
} ddef
/Tj
{
trj _ctm _pjss
} ddef
} def
/r1
{
/Tx
{
dup currentpoint 4 2 roll currentpoint gsave newpath moveto
tr _ctm _pss
grestore 3 1 roll moveto tr sp
} ddef
/Tj
{
dup currentpoint 4 2 roll currentpoint gsave newpath moveto
trj _ctm _pjss
grestore 3 1 roll moveto tr jsp
} ddef
} def
/To
{
pop _ctm currentmatrix pop
} def
/TO
{
iTe _ctm setmatrix newpath
} def
/Tp
{
pop _tm astore pop _ctm setmatrix
_tDict begin
/W
{
} def
/h
{
} def
} def
/TP
{
end
iTm 0 0 moveto
} def
/Tr
{
_render 3 le
{
currentpoint newpath moveto
} if
dup 8 eq
{
pop 0
}
{
dup 9 eq
{
pop 1
} if
} ifelse
dup /_render exch ddef
_renderStart exch get load exec
} def
/iTm
{
_ctm setmatrix _tm concat 0 _rise translate _hs 1 scale
} def
/Tm
{
_tm astore pop iTm 0 0 moveto
} def
/Td
{
_mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto
} def
/iTe
{
_render -1 eq
{
}
{
_renderEnd _render get dup null ne
{
load exec
}
{
pop
} ifelse
} ifelse
/_render -1 ddef
} def
/Ta
{
pop
} def
/Tf
{
dup 1000 div /_fScl exch ddef
%
selectfont
} def
/Tl
{
pop
0 exch _leading astore pop
} def
/Tt
{
pop
} def
/TW
{
3 npop
} def
/Tw
{
/_cx exch ddef
} def
/TC
{
3 npop
} def
/Tc
{
/_ax exch ddef
} def
/Ts
{
/_rise exch ddef
currentpoint
iTm
moveto
} def
/Ti
{
3 npop
} def
/Tz
{
100 div /_hs exch ddef
iTm
} def
/TA
{
pop
} def
/Tq
{
pop
} def
/Th
{
pop pop pop pop pop
} def
/TX
{
pop
} def
/Tk
{
exch pop _fScl mul neg 0 rmoveto
} def
/TK
{
2 npop
} def
/T*
{
_leading aload pop neg Td
} def
/T*-
{
_leading aload pop Td
} def
/T-
{
_ax neg 0 rmoveto
_hyphen Tx
} def
/T+
{
} def
/TR
{
_ctm currentmatrix pop
_tm astore pop
iTm 0 0 moveto
} def
/TS
{
currentfont 3 1 roll
/_Symbol_ _fScl 1000 mul selectfont
0 eq
{
Tx
}
{
Tj
} ifelse
setfont
} def
/Xb
{
pop pop
} def
/Tb /Xb load def
/Xe
{
pop pop pop pop
} def
/Te /Xe load def
/XB
{
} def
/TB /XB load def
currentdict readonly pop
end
setpacking
%%EndResource
%%BeginProcSet: Adobe_ColorImage_AI6 1.0 0
userdict /Adobe_ColorImage_AI6 known not
{
userdict /Adobe_ColorImage_AI6 17 dict put
} if
userdict /Adobe_ColorImage_AI6 get begin
/initialize
{
Adobe_ColorImage_AI6 begin
Adobe_ColorImage_AI6
{
dup type /arraytype eq
{
dup xcheck
{
bind
} if
} if
pop pop
} forall
} def
/terminate { end } def
currentdict /Adobe_ColorImage_AI6_Vars known not
{
/Adobe_ColorImage_AI6_Vars 14 dict def
} if
Adobe_ColorImage_AI6_Vars begin
/channelcount 0 def
/sourcecount 0 def
/sourcearray 4 array def
/plateindex -1 def
/XIMask 0 def
/XIBinary 0 def
/XIChannelCount 0 def
/XIBitsPerPixel 0 def
/XIImageHeight 0 def
/XIImageWidth 0 def
/XIImageMatrix null def
/XIBuffer null def
/XIDataProc null def
end
/WalkRGBString null def
/WalkCMYKString null def
/StuffRGBIntoGrayString null def
/RGBToGrayImageProc null def
/StuffCMYKIntoGrayString null def
/CMYKToGrayImageProc null def
/ColorImageCompositeEmulator null def
/SeparateCMYKImageProc null def
/FourEqual null def
/TestPlateIndex null def
currentdict /_colorimage known not
{
/colorimage where
{
/colorimage get /_colorimage exch def
}
{
/_colorimage null def
} ifelse
} if
/_currenttransfer systemdict /currenttransfer get def
/colorimage null def
/XI null def
/WalkRGBString
{
0 3 index
dup length 1 sub 0 3 3 -1 roll
{
3 getinterval { } forall
5 index exec
3 index
} for
5 { pop } repeat
} def
/WalkCMYKString
{
0 3 index
dup length 1 sub 0 4 3 -1 roll
{
4 getinterval { } forall
6 index exec
3 index
} for
5 { pop } repeat
} def
/StuffRGBIntoGrayString
{
.11 mul exch
.59 mul add exch
.3 mul add
cvi 3 copy put
pop 1 add
} def
/RGBToGrayImageProc
{
Adobe_ColorImage_AI6_Vars begin
sourcearray 0 get exec
dup length 3 idiv string
dup 3 1 roll
/StuffRGBIntoGrayString load exch
WalkRGBString
end
} def
/StuffCMYKIntoGrayString
{
exch .11 mul add
exch .59 mul add
exch .3 mul add
dup 255 gt { pop 255 } if
255 exch sub cvi 3 copy put
pop 1 add
} def
/CMYKToGrayImageProc
{
Adobe_ColorImage_AI6_Vars begin
sourcearray 0 get exec
dup length 4 idiv string
dup 3 1 roll
/StuffCMYKIntoGrayString load exch
WalkCMYKString
end
} def
/ColorImageCompositeEmulator
{
pop true eq
{
Adobe_ColorImage_AI6_Vars /sourcecount get 5 add { pop } repeat
}
{
Adobe_ColorImage_AI6_Vars /channelcount get 1 ne
{
Adobe_ColorImage_AI6_Vars begin
sourcearray 0 3 -1 roll put
channelcount 3 eq
{
/RGBToGrayImageProc
}
{
/CMYKToGrayImageProc
} ifelse
load
end
} if
image
} ifelse
} def
/SeparateCMYKImageProc
{
Adobe_ColorImage_AI6_Vars begin
sourcecount 0 ne
{
sourcearray plateindex get exec
}
{
sourcearray 0 get exec
dup length 4 idiv string
0 2 index
plateindex 4 2 index length 1 sub
{
get 255 exch sub
3 copy put pop 1 add
2 index
} for
pop pop exch pop
} ifelse
end
} def
/FourEqual
{
4 index ne
{
pop pop pop false
}
{
4 index ne
{
pop pop false
}
{
4 index ne
{
pop false
}
{
4 index eq
} ifelse
} ifelse
} ifelse
} def
/TestPlateIndex
{
Adobe_ColorImage_AI6_Vars begin
/plateindex -1 def
/setcmykcolor where
{
pop
gsave
1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub
grestore
1 0 0 0 FourEqual
{
/plateindex 0 def
}
{
0 1 0 0 FourEqual
{
/plateindex 1 def
}
{
0 0 1 0 FourEqual
{
/plateindex 2 def
}
{
0 0 0 1 FourEqual
{
/plateindex 3 def
}
{
0 0 0 0 FourEqual
{
/plateindex 5 def
} if
} ifelse
} ifelse
} ifelse
} ifelse
pop pop pop pop
} if
plateindex
end
} def
/colorimage
{
Adobe_ColorImage_AI6_Vars begin
/channelcount 1 index def
/sourcecount 2 index 1 eq { channelcount 1 sub } { 0 } ifelse def
4 sourcecount add index dup
8 eq exch 1 eq or not
end
{
/_colorimage load null ne
{
_colorimage
}
{
Adobe_ColorImage_AI6_Vars /sourcecount get
7 add { pop } repeat
} ifelse
}
{
dup 3 eq
TestPlateIndex
dup -1 eq exch 5 eq or or
{
/_colorimage load null eq
{
ColorImageCompositeEmulator
}
{
dup 1 eq
{
pop pop image
}
{
Adobe_ColorImage_AI6_Vars /plateindex get 5 eq
{
gsave
0 _currenttransfer exec
1 _currenttransfer exec
eq
{ 0 _currenttransfer exec 0.5 lt }
{ 0 _currenttransfer exec 1 _currenttransfer exec gt } ifelse
{ { pop 0 } } { { pop 1 } } ifelse
systemdict /settransfer get exec
} if
_colorimage
Adobe_ColorImage_AI6_Vars /plateindex get 5 eq
{
grestore
} if
} ifelse
} ifelse
}
{
dup 1 eq
{
pop pop
image
}
{
pop pop
Adobe_ColorImage_AI6_Vars begin
sourcecount -1 0
{
exch sourcearray 3 1 roll put
} for
/SeparateCMYKImageProc load
end
systemdict /image get exec
} ifelse
} ifelse
} ifelse
} def
/XI
{
Adobe_ColorImage_AI6_Vars begin
gsave
/XIMask exch 0 ne def
/XIBinary exch 0 ne def
pop
pop
/XIChannelCount exch def
/XIBitsPerPixel exch def
/XIImageHeight exch def
/XIImageWidth exch def
pop pop pop pop
/XIImageMatrix exch def
XIBitsPerPixel 1 eq
{
XIImageWidth 8 div ceiling cvi
}
{
XIImageWidth XIChannelCount mul
} ifelse
/XIBuffer exch string def
XIBinary
{
/XIDataProc { currentfile XIBuffer readstring pop } def
currentfile 128 string readline pop pop
}
{
/XIDataProc { currentfile XIBuffer readhexstring pop } def
} ifelse
0 0 moveto
XIImageMatrix concat
XIImageWidth XIImageHeight scale
XIMask
{
XIImageWidth XIImageHeight
false
[ XIImageWidth 0 0 XIImageHeight neg 0 0 ]
/XIDataProc load
/_lp /null ddef
_fc
/_lp /imagemask ddef
imagemask
}
{
XIImageWidth XIImageHeight
XIBitsPerPixel
[ XIImageWidth 0 0 XIImageHeight neg 0 0 ]
/XIDataProc load
XIChannelCount 1 eq
{
gsave
0 setgray
image
grestore
}
{
false
XIChannelCount
colorimage
} ifelse
} ifelse
grestore
end
} def
end
%%EndProcSet
%%BeginResource: procset Adobe_Illustrator_AI5 1.1 0
%%Title: (Adobe Illustrator (R) Version 5.0 Full Prolog)
%%Version: 1.1
%%CreationDate: (3/7/1994) ()
%%Copyright: ((C) 1987-1994 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_Illustrator_AI5_vars 81 dict dup begin
put
/_eo false def
/_lp /none def
/_pf
{
} def
/_ps
{
} def
/_psf
{
} def
/_pss
{
} def
/_pjsf
{
} def
/_pjss
{
} def
/_pola 0 def
/_doClip 0 def
/cf currentflat def
/_tm matrix def
/_renderStart
[
/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0
] def
/_renderEnd
[
null null null null /i1 /i1 /i1 /i1
] def
/_render -1 def
/_rise 0 def
/_ax 0 def
/_ay 0 def
/_cx 0 def
/_cy 0 def
/_leading
[
0 0
] def
/_ctm matrix def
/_mtx matrix def
/_sp 16#020 def
/_hyphen (-) def
/_fScl 0 def
/_cnt 0 def
/_hs 1 def
/_nativeEncoding 0 def
/_useNativeEncoding 0 def
/_tempEncode 0 def
/_pntr 0 def
/_tDict 2 dict def
/_wv 0 def
/Tx
{
} def
/Tj
{
} def
/CRender
{
} def
/_AI3_savepage
{
} def
/_gf null def
/_cf 4 array def
/_if null def
/_of false def
/_fc
{
} def
/_gs null def
/_cs 4 array def
/_is null def
/_os false def
/_sc
{
} def
/_pd 1 dict def
/_ed 15 dict def
/_pm matrix def
/_fm null def
/_fd null def
/_fdd null def
/_sm null def
/_sd null def
/_sdd null def
/_i null def
/discardSave null def
/buffer 256 string def
/beginString null def
/endString null def
/endStringLength null def
/layerCnt 1 def
/layerCount 1 def
/perCent (%) 0 get def
/perCentSeen? false def
/newBuff null def
/newBuffButFirst null def
/newBuffLast null def
/clipForward? false def
end
userdict /Adobe_Illustrator_AI5 known not {
userdict /Adobe_Illustrator_AI5 91 dict put
} if
userdict /Adobe_Illustrator_AI5 get begin
/initialize
{
Adobe_Illustrator_AI5 dup begin
Adobe_Illustrator_AI5_vars begin
discardDict
{
bind pop pop
} forall
dup /nc get begin
{
dup xcheck 1 index type /operatortype ne and
{
bind
} if
pop pop
} forall
end
newpath
} def
/terminate
{
end
end
} def
/_
null def
/ddef
{
Adobe_Illustrator_AI5_vars 3 1 roll put
} def
/xput
{
dup load dup length exch maxlength eq
{
dup dup load dup
length 2 mul dict copy def
} if
load begin
def
end
} def
/npop
{
{
pop
} repeat
} def
/sw
{
dup length exch stringwidth
exch 5 -1 roll 3 index mul add
4 1 roll 3 1 roll mul add
} def
/swj
{
dup 4 1 roll
dup length exch stringwidth
exch 5 -1 roll 3 index mul add
4 1 roll 3 1 roll mul add
6 2 roll /_cnt 0 ddef
{
1 index eq
{
/_cnt _cnt 1 add ddef
} if
} forall
pop
exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop
} def
/ss
{
4 1 roll
{
2 npop
(0) exch 2 copy 0 exch put pop
gsave
false charpath currentpoint
4 index setmatrix
stroke
grestore
moveto
2 copy rmoveto
} exch cshow
3 npop
} def
/jss
{
4 1 roll
{
2 npop
(0) exch 2 copy 0 exch put
gsave
_sp eq
{
exch 6 index 6 index 6 index 5 -1 roll widthshow
currentpoint
}
{
false charpath currentpoint
4 index setmatrix stroke
} ifelse
grestore
moveto
2 copy rmoveto
} exch cshow
6 npop
} def
/sp
{
{
2 npop (0) exch
2 copy 0 exch put pop
false charpath
2 copy rmoveto
} exch cshow
2 npop
} def
/jsp
{
{
2 npop
(0) exch 2 copy 0 exch put
_sp eq
{
exch 5 index 5 index 5 index 5 -1 roll widthshow
}
{
false charpath
} ifelse
2 copy rmoveto
} exch cshow
5 npop
} def
/pl
{
transform
0.25 sub round 0.25 add exch
0.25 sub round 0.25 add exch
itransform
} def
/setstrokeadjust where
{
pop true setstrokeadjust
/c
{
curveto
} def
/C
/c load def
/v
{
currentpoint 6 2 roll curveto
} def
/V
/v load def
/y
{
2 copy curveto
} def
/Y
/y load def
/l
{
lineto
} def
/L
/l load def
/m
{
moveto
} def
}
{
/c
{
pl curveto
} def
/C
/c load def
/v
{
currentpoint 6 2 roll pl curveto
} def
/V
/v load def
/y
{
pl 2 copy curveto
} def
/Y
/y load def
/l
{
pl lineto
} def
/L
/l load def
/m
{
pl moveto
} def
} ifelse
/d
{
setdash
} def
/cf
{
} def
/i
{
dup 0 eq
{
pop cf
} if
setflat
} def
/j
{
setlinejoin
} def
/J
{
setlinecap
} def
/M
{
setmiterlimit
} def
/w
{
setlinewidth
} def
/XR
{
0 ne
/_eo exch ddef
} def
/H
{
} def
/h
{
closepath
} def
/N
{
_pola 0 eq
{
_doClip 1 eq
{
_eo {eoclip} {clip} ifelse /_doClip 0 ddef
} if
newpath
}
{
/CRender
{
N
} ddef
} ifelse
} def
/n
{
N
} def
/F
{
_pola 0 eq
{
_doClip 1 eq
{
gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc
/_doClip 0 ddef
}
{
_pf
} ifelse
}
{
/CRender
{
F
} ddef
} ifelse
} def
/f
{
closepath
F
} def
/S
{
_pola 0 eq
{
_doClip 1 eq
{
gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc
/_doClip 0 ddef
}
{
_ps
} ifelse
}
{
/CRender
{
S
} ddef
} ifelse
} def
/s
{
closepath
S
} def
/B
{
_pola 0 eq
{
_doClip 1 eq
gsave F grestore
{
gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc
/_doClip 0 ddef
}
{
S
} ifelse
}
{
/CRender
{
B
} ddef
} ifelse
} def
/b
{
closepath
B
} def
/W
{
/_doClip 1 ddef
} def
/*
{
count 0 ne
{
dup type /stringtype eq
{
pop
} if
} if
newpath
} def
/u
{
} def
/U
{
} def
/q
{
_pola 0 eq
{
gsave
} if
} def
/Q
{
_pola 0 eq
{
grestore
} if
} def
/*u
{
_pola 1 add /_pola exch ddef
} def
/*U
{
_pola 1 sub /_pola exch ddef
_pola 0 eq
{
CRender
} if
} def
/D
{
pop
} def
/*w
{
} def
/*W
{
} def
/`
{
/_i save ddef
clipForward?
{
nulldevice
} if
6 1 roll 4 npop
concat pop
userdict begin
/showpage
{
} def
0 setgray
0 setlinecap
1 setlinewidth
0 setlinejoin
10 setmiterlimit
[] 0 setdash
/setstrokeadjust where {pop false setstrokeadjust} if
newpath
0 setgray
false setoverprint
} def
/~
{
end
_i restore
} def
/O
{
0 ne
/_of exch ddef
/_lp /none ddef
} def
/R
{
0 ne
/_os exch ddef
/_lp /none ddef
} def
/g
{
/_gf exch ddef
/_fc
{
_lp /fill ne
{
_of setoverprint
_gf setgray
/_lp /fill ddef
} if
} ddef
/_pf
{
_fc
_eo {eofill} {fill} ifelse
} ddef
/_psf
{
_fc
ashow
} ddef
/_pjsf
{
_fc
awidthshow
} ddef
/_lp /none ddef
} def
/G
{
/_gs exch ddef
/_sc
{
_lp /stroke ne
{
_os setoverprint
_gs setgray
/_lp /stroke ddef
} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
/k
{
_cf astore pop
/_fc
{
_lp /fill ne
{
_of setoverprint
_cf aload pop setcmykcolor
/_lp /fill ddef
} if
} ddef
/_pf
{
_fc
_eo {eofill} {fill} ifelse
} ddef
/_psf
{
_fc
ashow
} ddef
/_pjsf
{
_fc
awidthshow
} ddef
/_lp /none ddef
} def
/K
{
_cs astore pop
/_sc
{
_lp /stroke ne
{
_os setoverprint
_cs aload pop setcmykcolor
/_lp /stroke ddef
} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
/x
{
/_gf exch ddef
findcmykcustomcolor
/_if exch ddef
/_fc
{
_lp /fill ne
{
_of setoverprint
_if _gf 1 exch sub setcustomcolor
/_lp /fill ddef
} if
} ddef
/_pf
{
_fc
_eo {eofill} {fill} ifelse
} ddef
/_psf
{
_fc
ashow
} ddef
/_pjsf
{
_fc
awidthshow
} ddef
/_lp /none ddef
} def
/X
{
/_gs exch ddef
findcmykcustomcolor
/_is exch ddef
/_sc
{
_lp /stroke ne
{
_os setoverprint
_is _gs 1 exch sub setcustomcolor
/_lp /stroke ddef
} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
/A
{
pop
} def
/annotatepage
{
userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse
} def
/XT {
pop pop
} def
/discard
{
save /discardSave exch store
discardDict begin
/endString exch store
gt38?
{
2 add
} if
load
stopped
pop
end
discardSave restore
} bind def
userdict /discardDict 7 dict dup begin
put
/pre38Initialize
{
/endStringLength endString length store
/newBuff buffer 0 endStringLength getinterval store
/newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store
/newBuffLast newBuff endStringLength 1 sub 1 getinterval store
} def
/shiftBuffer
{
newBuff 0 newBuffButFirst putinterval
newBuffLast 0
currentfile read not
{
stop
} if
put
} def
0
{
pre38Initialize
mark
currentfile newBuff readstring exch pop
{
{
newBuff endString eq
{
cleartomark stop
} if
shiftBuffer
} loop
}
{
stop
} ifelse
} def
1
{
pre38Initialize
/beginString exch store
mark
currentfile newBuff readstring exch pop
{
{
newBuff beginString eq
{
/layerCount dup load 1 add store
}
{
newBuff endString eq
{
/layerCount dup load 1 sub store
layerCount 0 eq
{
cleartomark stop
} if
} if
} ifelse
shiftBuffer
} loop
} if
} def
2
{
mark
{
currentfile buffer readline not
{
stop
} if
endString eq
{
cleartomark stop
} if
} loop
} def
3
{
/beginString exch store
/layerCnt 1 store
mark
{
currentfile buffer readline not
{
stop
} if
dup beginString eq
{
pop /layerCnt dup load 1 add store
}
{
endString eq
{
layerCnt 1 eq
{
cleartomark stop
}
{
/layerCnt dup load 1 sub store
} ifelse
} if
} ifelse
} loop
} def
end
userdict /clipRenderOff 15 dict dup begin
put
{
/n /N /s /S /f /F /b /B
}
{
{
_doClip 1 eq
{
/_doClip 0 ddef _eo {eoclip} {clip} ifelse
} if
newpath
} def
} forall
/Tr /pop load def
/Bb {} def
/BB /pop load def
/Bg {12 npop} def
/Bm {6 npop} def
/Bc /Bm load def
/Bh {4 npop} def
end
/Lb
{
4 npop
6 1 roll
pop
4 1 roll
pop pop pop
0 eq
{
0 eq
{
(%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard
}
{
/clipForward? true def
/Tx /pop load def
/Tj /pop load def
currentdict end clipRenderOff begin begin
} ifelse
}
{
0 eq
{
save /discardSave exch store
} if
} ifelse
} bind def
/LB
{
discardSave dup null ne
{
restore
}
{
pop
clipForward?
{
currentdict
end
end
begin
/clipForward? false ddef
} if
} ifelse
} bind def
/Pb
{
pop pop
0 (%AI5_EndPalette) discard
} bind def
/Np
{
0 (%AI5_End_NonPrinting--) discard
} bind def
/Ln /pop load def
/Ap
/pop load def
/Ar
{
72 exch div
0 dtransform dup mul exch dup mul add sqrt
dup 1 lt
{
pop 1
} if
setflat
} def
/Mb
{
q
} def
/Md
{
} def
/MB
{
Q
} def
/nc 3 dict def
nc begin
/setgray
{
pop
} bind def
/setcmykcolor
{
4 npop
} bind def
/setcustomcolor
{
2 npop
} bind def
currentdict readonly pop
end
end
setpacking
%%EndResource
%%EndProlog
%%BeginSetup
%%IncludeFont: Symbol
%%IncludeFont: Times-Italic
%%IncludeFont: Times-Roman
Adobe_level2_AI5 /initialize get exec
Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec
Adobe_ColorImage_AI6 /initialize get exec
Adobe_Illustrator_AI5 /initialize get exec
[
39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis
/Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute
/egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde
/oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex
/udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls
/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash
/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef
/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash
/questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef
/guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe
/endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide
/.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright
/fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand
/Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex
/Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex
/Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla
/hungarumlaut/ogonek/caron
TE
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%AI3_EndEncoding TrueType
%AI3_BeginEncoding: _Times-Roman Times-Roman
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\begin{document}
\title{A tutorial on KAM theory}
% Information for first author
\author{Rafael de la Llave}
\address{Department of Mathematics, The University of Texas at Austin,
Austin, TX 78712-1082}
\email{llave@math.utexas.edu}
\subjclass{Primary 37J40; }
\keywords{KAM theory, stability, Perturbation theory,quasiperiodic orbits,Hamiltonian systems}
\begin{abstract}
This is a tutorial on some of the main ideas in KAM theory.
The goal is to present the background and to explain and
compare somewhat informally some of the main methods of proof.
It is an expanded version of the lectures given by the author
in the Summer Research Institute on {\sl Smooth Ergodic Theory}
Seattle, 1999. The style is pedagogical and expository
and it only aims to be an introduction to the primary literature.
It does not aim to be a systematic survey nor to present
full proofs.
\end{abstract}
\maketitle
\tableofcontents
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
The goal of these lectures is to present
an introduction to some of the main ideas involved in
KAM theory on the persistence of quasiperiodic motions under
perturbations. The name comes from the initials of
A. N. Kolmogorov, V. I. Arnol'd and J. Moser who initiated
the theory. See \cite{Kolmogorov79}, \cite{Arnold63a},
\cite{Arnold63b}, \cite{Moser62}, \cite{Moser66a},
\cite{Moser66b} for the original papers.
By now, it is a full fledged theory and it
provides a systematic tool for the analysis of many
dynamical systems and it also has relations with
other areas of analysis.
The conclusions of the theory are, roughly, that
in $C^k$ -- $k$ rather high depending on the dimension --
open sets of
of dynamical systems satisfying some geometric properties
-- e.g. Hamiltonian, volume preserving, reversible, etc. --
there are sets of positive measure covered by
invariant tori. In particular, since sets with
a positive measure of invariant tori is incompatible
with ergodicity, we conclude that for the systems
mentioned above, ergodicity
cannot be a $C^k$ generic property
\cite{MarkusM74}.
Of course, the existence of
the quasiperiodic orbits, has many other
consequences besides preventing ergodicity.
The invariant tori are
important landmarks
that guide the motion.
Besides its applications to mechanics, dynamical systems
and ergodic theory, KAM theory
has grown enormously and has very interesting ramifications
in dynamical systems and in Analysis.
For example, averaging theory gives rise to
estimates for very long times valid for all
initial conditions, one can use partially
hyperbolic tori to show existence of orbits that escape.
On the analytical side, the theory
leads to functional analysis methods that can be used to
solve a variety of functional equations, many of which have
interest in ergodic theory and in related disciplines
such as differential geometry.
There already exist excellent surveys,
systematic expositions
and tutorials of KAM theory.
We quote in chronological order:
\cite{Arnold63b},
\cite{Moser66a,Moser66b,Moser73},
\cite{ArnoldA68},
\cite{Russman70},
\cite{Russman72},
\cite{Zehnder75,Zehnder76},
\cite{Douady82a},
\cite{Bost86},
\cite{Salamon86},
\cite{Poschel},
\cite{Poschel92}
\cite{Yoccoz92b},
\cite{ArnoldKN93},
\cite{Llave93}.
\cite{BroerHS96},
\cite{Gallavotti1},
\cite{Gallavotti2},
\cite{Wayne96},
\cite{Russmann98},
and
\cite{Marmi99}.
Hence, one has to justify the effort in writing
and reading yet another exposition.
I decided that each of the surveys above has picked up
a particular point of view and tried to either present
a large part of KAM theory from this point of
view or to provide a particularly enlightening
example.
Given the high quality of all (but one) of
the above surveys and tutorials, there seems to be little
point in trying to achieve the same goals.
Therefore, rather than presenting
a point of view with full
proofs, this tutorial will have only the more
modest goal of summarizing some of the
main ideas entering into KAM theory and
describing and comparing
the main points of view.
Therefore,
it is not a substitute for the
full papers we reference.
One of the disadvantages of covering
such wide ground is that the presentation will have to be
sketchy at some points. Hopefully, we have flagged
a good fraction of these sketchy points and
referred to the relevant literature.
I would be happy if these lectures provide a road map
(necessarily omitting important details)
of a fraction of the literature
that encourages somebody to enter into the field. Needless to say,
this is not a survey and we have not made any attempt to
be systematic nor to reach the forefront of research.
It should be kept in mind that KAM theory has
experienced spectacular pro\-gress in recent years
and that it is a very active area of research.
\subsection{Some recent developments} \label{newdevelopments}
Let me mention some of these new developments (in no particular order and,
with no claim of completeness of the list and omitting classical results
-- i.e. more that 15 years old --).
They will not be covered in the lectures, which will be
concerned only with the most classical results.
The novice that is reading the paper to
get initiated to KAM theory is encouraged to skip it for the moment
and only come back to it as suggestions for future reading.
Of course, the experts will notice many omissions. The only point
we are trying to make is that the theory is still finding exciting
results and that there is work to do.
\begin{itemize}
\item The ``lack of parameters" which was
considered inaccessible
has been solved very elegantly
\cite{JorbaS92},
\cite{Eliasson88a}.
(See \cite{BroerHS96} for a recent survey, and also
\cite{BroerHS96b}
\cite{Sevryuk99}.)
This has lead to remarkable progress in the
existence of lower dimensional tori, specially elliptic
tori -- a theory of hyperbolic tori has been known for
a long time --
(see e.g. \cite{JorbaV97a}, \cite{JorbaV97b}.)
\item As a corollary of this, one can get a
reasonable KAM theory for volume preserving systems
getting tori of codimension one. Hence blocking diffusion in many
problems in hydrodynamics, etc.
(See \cite{ChengS89}, \cite{BroerHS96}, \cite{DelshamsL90},
\cite{Xia92}, \cite{Yoccoz92b}.)
\item
The
KAM theory for infinite dimensional systems has made remarkable progress.
Note that in infinite dimensional systems, the most
interesting tori are of lower dimension than the number of
degrees of freedom.
The subject of infinite dimensional KAM by itself
would require a review of its own longer than these
notes.
We just refer to \cite{CraigW93}, \cite{CraigW94},
\cite{Poschel96},
\cite{Bourgain95},
\cite{Bourgain00},
and \cite{Kuksin93} as representative references,
where the interested reader can find further references.
\item Many systems in applications -- e.g in statistical
mechanics -- have the structure that they consists of
arrays of systems connected by local couplings.
\footnote{
These systems, under the name of {\sl coupled lattice maps}
have also been the subject of very intense research
when they have hyperbolicity properties, in some sense
opposite to the situation considered in KAM theory}
For these systems one can take advantage of this
structure and develop a more efficient KAM theory
than the simple application of the general results.
\cite{Wayne84}, \cite{Poschel90}
\cite{FrolichSW86}. Other KAM
methods for these systems are
developed in \cite{AlbaneseFS88a}
and \cite{albaneseFS88b}, \cite{AlbaneseF91}
which consider the existence of periodic solutions.
See also \cite{Wayne86} for
an Nekhoroshev theorem for these systems.
Conjectures and preliminary estimates (a challenge for
rigorous proofs) on these systems can
be found in
\cite{BenettinGG85a},
\cite{BenettinGG85b}, \cite{HaroL00},\cite{CassettiCPC97}.
\item The non-degeneracy conditions needed for KAM theorems
have been greatly weakened \cite{Russmann90}, \cite{ChengS94},
\cite{Russmann98}.
See also \cite{BroerHS96}, \cite{BroerHS96b},\cite{Sevryuk95}, \cite{Sevryuk96}.
\item Modern techniques of PDE's such as viscosity
solutions have been used to study the
Hamilton-Jacobi equation \cite{Lions82}, \cite{CrandallL83},
\cite{CrandallEL84},
leading to a weak version of KAM theory
that has deep relations with Aubry-Mather theory
\cite{Fathi97a}, \cite{Fathi97b}.
\item There has been
quite spectacular progress in the
problem of {\sl reducibility} of linear equations
with quasiperiodic coefficients
(That is, the study whether an equation with the form
$\dot x = A( \phi + \omega t) x $
where $A \rightarrow \torus^d \M_{n \times n}$ and
$\omega \in \real^d$ is an irrational vector.
can be transformed into constant coefficients.
After the original work of \cite{DinaburgS75},
two important recent developments were
\cite{MoserP84} which introduced the deep idea of
using transformations which are not close to the identity
to eliminate small terms and \cite{Rychlik92} which introduced
a renormalization mechanism.
After that, many more new important refinements
were introduced in several works (one needs to
find ways to combine perturbative steps with non-perturbative
ones). This is still a very active area and progress is
being made constantly. We refer to the lectures of
prof. Eliasson in this volume for up to date references.
See also \cite{Eliasson98},\cite{Krikorian99a}.
\item The problem of reducibility is related to the problem of
existence of pure point spectrum of one-dimensional
Schr\"odinger operators with quasiperiodic coefficients.
This area has experienced quite significant progress.
Besides some of the papers mentioned in the previous
paragraphs, let us mention
\cite{ChulaevskyS91},
\cite{FrolichSW90},
\cite{Eliasson97}.
\item For Schr\"odinger operators in higher dimensions
with random or
quasi-periodic potential the theory of localization also
has advanced greatly thanks to a multi-scale analysis which is
quite reminiscent of KAM theory \cite{FrolichS83},\cite{FrolichS84}.
Indeed, this analogy has been pursued quite fruitfully.
\cite{Albanese93}.
\item Even if the symplectic forms that appear in mechanical
systems
admit a primitive (see later in Section \ref{Geometric_structures}),
there are other symplectic forms
without this feature.
For such forms without a primitive,
one has the possibility of finding
persistent tori of more dimension than the degrees
of freedom. This has important consequences and
leads to very interesting examples in ergodic theory.
See \cite{Yoccoz92b}, \cite{Herman91}
(See also \cite{Parasyuk84}, \cite{Parasyuk89},
\cite{FuzhongY98}.)
\item KAM methods have been extended to
elliptic PDE's -- they are not evolution equations.
The role of time in KAM has been taken by
spatial variables. (See \cite{Kozlov83}, \cite{Moser88}, \cite{Moser95}.)
This has also been related to a variational
structure of the equations \cite{Moser86b}, \cite{Bangert89}.
\item There are some proofs of KAM type theorems based on
different principles, notably renormalization group,
\cite{BricmontGK99}, \cite{Koch}, \cite{Kosygin91}.
This is perhaps related to some recent proofs that
do not even use Fourier analysis
\cite{KhaninS86},
\cite{SinaiK87}, \cite{SinaiK89},
\cite{KatznelsonO93}, \cite{KatznelsonO89a}, \cite{KatznelsonO89b},
\cite{Stark88}, \cite{Haydn90}
\cite{Stirnemann94}.
\item More interestingly, renormalization group has
been used to describe the breakdown of
invariant circles, starting with \cite{McKay82} -- which
includes a beautiful picture in terms of fixed points
and manifolds of operators and makes very
detailed predictions about scalings at breakdown -- or \cite{EscandeD81},
which contains a simpler approach that gives less
detailed predictions.
Much of what is known at this level remains
at the level of numerical well founded conjectures.
Indeed, there are still quite important issues that
are not even known at this level.
Among the rigorous work in this area, we
mention \cite{Stirnemann93}, \cite{Stirnemann97}.
\item KAM theory has started to become a tool of applied
mathematics with the advent of constructive methods
to asses the reliability of numerical computations
\cite{CellettiC95}, \cite{LlaveR90}, \cite{Schmidt95}, \cite{Jorba99}.
\item For some special cases of KAM theory, there has
also been very important progress examining the limits of validity;
the role of the arithmetic
conditions has been clarified for complex mappings -- specially
quadratic -- \cite{Yoccoz95}.
See also \cite{Perezmarco00}.
The study
of the radius of convergence of the linearization
in the same mappings
\cite{MarmiMY97} has also been quite well understood.
In some twist mappings, there has been a very significant advances in the
study of non-existence of tori
\cite{Mather88}, \cite{McKayP85},
\cite{Jungreis91}. The domains of convergence
of the perturbative expansions have been analyzed using
tools similar to those used for analytic complex mappings starting
in \cite{Davie94} -- a map which has features between
those of a complex analytic map and those of
a twist map -- and then in \cite{MarmiS92}, \cite{BerrettiM95},
\cite{BerrettiG98}.
\item
Two different techniques to study
quasiperiodic orbits on twist maps
are the variational
methods of Mather \cite{MatherF91} and the
renormalization group \cite{Koch};
In many cases, these theories have ranges of
validity much greater than those covered by
KAM theory and, therefore provide some glimpse into
what happens at the breakdown of KAM theory.
\item There has been great progress
in using ``direct methods'', which are based on
writing a perturbative expansion and showing it
converges by studying more deeply the
structure of small denominators.
In the study of iterations of
analytic functions, these methods led
to the original proof of
Siegel \cite{Siegel42}, which was the first problem in
which small denominators were understood.
They were also used in the first proof
of the optimal arithmetic conditions
\cite{Brjuno71}.
In the study of Lindstedt series (see Section \ref{linstedt}),
the proof of convergence by exhibiting
explicitly cancellations of the series
was accomplished in \cite{Eliasson96}
(the preprint circulated much earlier).
The proof of the convergence of the
Lindstedt series in \cite{Eliasson96} is much more subtle
than that of \cite{Siegel42}.
Contrary to the terms in the expansions considered
in \cite{Siegel42}, the terms in the
Lindstedt series do grow very fast and one
cannot establish convergence by just bounding sizes
but one needs to exhibit cancellations
in the terms.
Expositions and simplifications of this work
relating it also to techniques of
perturbative Quantum Field Theory
can be found in \cite{Gallavotti94b},
\cite{GallavottiG95},
\cite{ChierchiaF94} and extensions
to some PDE's
in \cite{ChierchiaF96}.
Direct methods not only provide
alternative proofs of known facts,
but also have been used to prove several
results, which at the moment do not seem
to have proofs using rapidly convergent methods.
To my knowledge, the following results
established using direct methods do
not have rapidly convergent proofs:
The existence of
some invariant manifolds contained in
center manifolds in \cite{Poschel86}
was proved using cancellations similar to those
of Siegel.
It seems that there are no rapidly convergent
proofs of these results
(however, see \cite{Stolovitch94a},
\cite{Stolovitch94b} which solve a very related
problem.)
The deeper cancellations of
\cite{Eliasson96} have been used to
give a proof of the Gallavotti conjectures
(which imply, among other consequences, the
amusing result that an analytic Hamiltonian
near an elliptic fixed point is the sum of
two integrable systems -- of course integrated in
different coordinates.)
\cite{Eliasson88}
and to prove the
existence of quasi-flat intersections in \cite{Gallavotti94}.
A problem that remains open is the fact that the Lindstedt
series for lower dimensional KAM tori involve less
small divisors conditions than the KAM proof.
(See \cite{JorbaLZ00} for a discussion of this problem.)
\item Subjects closely related to KAM theory such
as averaging and Nekhoroshev theory have also
experienced a great deal of development.
\item
Even if this is somewhat out of the line of
topics to be discussed here, we note
that related fields such as averaging theory
and Nekhoroshev estimates has also experienced
very important
developments. Let us just mention very
quickly: An elegant proof of the
theorem based on approximation by periodic orbits
\cite{Lochak92}, the proof
of what are conjectured to be the optimal exponents
\cite{LochakN92}, \cite{Poschel93},
\cite{DelshamsG96} -- the later paper contains a
unified point of view for KAM and Nekhoroshev theorems --
and the proof of Nekhoroshev estimates in a
neighborhood of an elliptic fixed point
\cite{GuzzoFB98b}, \cite{FassoGB98a},
\cite{Niederman98}, \cite{Poschel99}.
In a more innovative direction, Nekhoroshev
type theorems for PDE's have been
established \cite{BambusiN98}, \cite{Nekhoroshev99},
\cite{Bambusi99a}, \cite{Bambusi99b}.
\item The list could (perhaps should) be continued, with other topics
that are related to KAM theory and connecting it to
other theories of mechanics,
such as averaging theory, Aubry-Mather theory, quantum versions of
KAM theory, rigidity theory, exponential asymptotics or
Arnol'd diffusion and many others which are not even mentioned
mainly because of the ignorance of the lecturer, which he is the
first to regret.
\end{itemize}
Needless to say in this tutorial, we cannot hope to do
justice to all the topics above.
(Indeed, I have little hope that the above
list of topics and references is complete.)
The only goal is to provide an entry point to the main
ideas that will need to be read from the literature
and, possibly, to convey some of the excitement and the beauty
of this area of research.
Clearly, I cannot (and I do not) make any claim of
originality or completeness. This is not
a systematic survey of
topics of current research. The modest goal I set
set for these notes is to
help some readers to get started in the
beautiful and active subject of KAM theory
by giving a crude road map. I just
hope that the many deficiencies of
this tutorial will incense somebody into writing a proper review or
a better tutorial.
In the mean time, I will be happy to receive
comments, corrections and suggestions for improvement
of this tutorial
which I will make available electronically.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Some motivating examples}
\mar{linstedt}
\subsection{Lindstedt series for twist maps} \label{linstedt}
One of the original motivations of KAM theory was the study of
quasi-periodic solutions of Hamiltonian systems.
In this Section we will cover some elementary and well-known examples.
One particularly motivating example is the so-called {\em standard map}.
This is a map from $\real \times \torus$ to itself.
We denote the real coordinate by $p$ and the angle
one by $q$. Denoting by $p_n$, $q_n$ the values of
these coordinates at the discrete time $n$, the
map can be written as:
\mar{standardhamiltonian}
\begin{equation}\label{standardhamiltonian}
\begin{array}{rcl}
p_{n+1} & = & p_n - \ep V' (q_n)\\
q_{n+1} & = & (q_n + p_{n+1}) \mod 1 ,
\end{array}
\end{equation}
where $V(x) = V(x+1)$ is a smooth
(for our purposes in this Section, analytic) function.
We will also use a more explicit
expression for the map.
\mar{standardmap}
\begin{equation}\label{standardmap}
T_\ep(p,q) = \left( p - \ep V'(q), \, q + p - \ep V'(q) \right) \ .
\end{equation}
Substituting the expression for $p_{n+1}$ given in the second
equation of \eqref{standardhamiltonian} into the first,
we see that the system \eqref{standardhamiltonian} is
equivalent to the second order equation.
\mar{standardlagrangian}
\begin{equation}
\label{standardlagrangian}
q_{n+1} + q_{n-1} - 2q_n = -\ep V' (q_n) \ ,
\end{equation}
The first, ``Hamiltonian'', formulation
\eqref{standardhamiltonian} appears naturally in some mechanical
systems (e.g., the kicked pendulum).
The second, ``Lagrangian'', one \eqref{standardlagrangian}
appears naturally from a variational principle,
namely, it is equivalent to
the equations
\begin{equation}\label{EulerLagrange}
\partial\L/\partial q_n =0
\end{equation}
with
\mar{Frenkel-Kontorova}
\begin{equation}\label{Frenkel-Kontorova}
\L(q) = - \sum_n \left[\frac12 (q_{n+1} - q_n -a)^2 + \ep V(q_n)\right] \ .
\end{equation}
The equations \eqref{EulerLagrange} -- often called
Euler-Lagrange equations --
express that $\{q_n\}$ is a critical point for the action
\eqref{Frenkel-Kontorova}.
The model \eqref{Frenkel-Kontorova} has appeared in solid state physics
under the name {\em Frenkel-Kontorova model}. (See e.g. \cite{AubryD83}.)
One physical interpretation (not the only
possible one) that has lead to
many heuristic insights
is that $q_n$ is the position of the
$n^{\rm th}$ atom in a chain.
These atoms interact with their nearest neighbors
by the quadratic potential energy $\frac12 (q_{n+1} -q_n -a)^2$
(corresponding to springs connecting the nearest neighbors)
and with a substratum by the potential energy $\ep V(q_n)$.
The parameter $a$ is the equilibrium length of each spring.
Note that $a$ drops from the equilibrium equations
\eqref{standardlagrangian} but affects which among all the equilibria
corresponds to a minimum of the energy.
Another interpretation, of more interest for the
theme of these lectures, is
that $q_n$ are the positions at consecutive
times of a one-degree of freedom twist map.
The general term in the sum
$S(q_{n+1},q_n)$
are the
generating functions of the map.
(See Section \ref{generating}.)
Then, the Euler-Lagrange equations for
critical points of the functional are
equivalent to the sequence $\{q_n\}$ being the
projection of an orbit.
The first formulation \eqref{standardhamiltonian}
is area preserving whenever $V'$ is a periodic
function of the cylinder -- not necessarily the derivative
of a periodic function
(i.e., the Jacobian of the transformation
$(p_n, q_n) \mapsto (p_{n+1}, q_{n+1})$ is equal to~1).
When, as we have indicated, $V'$ is indeed the derivative
of a periodic function, then the map
is {\sl exact}, a concept that we will discuss in greater
detail in Section \ref{Geometric_structures} and that has
great importance for KAM theory.
If we look at the map \eqref{standardhamiltonian}
for $\ep=0$, we note that it becomes
\mar{integrable}
\begin{equation}\label{integrable}
\begin{array}{rcl}
p_{n+1} & = & p_n\\
q_{n+1} & = & q_n + p_n \ ,
\end{array}
\end{equation}
so that the ``horizontal'' circles $\{p_n={\rm const},\,n\in\zed\}$
in the cylinder are preserved and the motion of each $q_n$
in each circle is a rigid rotation that is faster
in the circles with larger $p_n$.
Note that when $p_0$ is an irrational number,
a classical elementary theorem in number theory shows
that the orbit is dense on the circle.
(A deeper theorem due to Weyl shows that it is actually
equidistributed in the circle.)
We are interested in finding whether, when we turn on the perturbation
$\ep$, some of this behavior persists.
More concretely, we are interested in knowing whether there are
quasi-periodic orbits that persist and that fill a circle densely.
Problems that are qualitatively similar to \eqref{standardhamiltonian}
appear in celestial mechanics \cite{SiegelM95}
and the role of these quasi-periodic orbits
have been appreciated for many years.
One can already find a rather systematic study in \cite{Poincare}
and the treatment there refers to many older works.
We note that the existence of quasi-periodic orbits
is hopeless if one allows general perturbations of
\eqref{integrable}.
For example, if we take a map of the form
\mar{example1}
\begin{equation}\label{example1}
\begin{array}{rcl}
p_{n+1} & = & p_n - \ep p_n\\
q_{n+1} & = & q_n + p_{n+1} \ ,
\end{array}
\end{equation}
we see that
applying repeatedly
\eqref{example1},
we have
$$
p_n = (1 -\ep)^n p_0
$$
so that, when $0 < \ep < 2$,
all orbits concentrate on the very small set $p=0$ and that
we get at most only one frequency. When $\ep < 0$ or $\ep > 2$,
all the orbits except those in $p = 0$, blow up to infinity.
Hence, we can have maps with radically different dynamical
behavior by making arbitrarily small perturbations.
More subtly, the orbits of
\mar{example2}
\begin{equation}\label{example2}
\begin{array}{rcl}
p_{n+1} & = & p_n +\ep\\
q_{n+1} & = & q_n + p_{n+1}
\end{array}
\end{equation}
escape towards infinity and never come back to themselves
(in particular, can never be quasi-periodic).
The first example is not area preserving and the motion is concentrated
in a smaller area (in particular, it does not come back to itself).
The second example is area preserving but
has non-zero ``flux''.
\mar{fig1}
\begin{figure}
\centerline{\epsfysize=1.5truein\epsfbox{llave-fig1.eps}}
\label{fig1}
\caption{The flux is the oriented area between a circle and its image.}
\end{figure}
\begin{definition}\label{flux}
The ``flux'' of an area preserving map $T$ of the cylinder
is defined as follows:
given a continuous circle $\gamma$ on the cylinder,
the flux of $T$ is the oriented area between $T(\gamma)$,
the image of the circle, and~$\gamma$ ---
see Figure 1.
\end{definition}
The fact that the map is area preserving implies
easily that this flux is
independent of the circle (hence it is an invariant of the map).
Clearly, if the map $T$ had a continuous invariant circle,
the flux should be zero,
so we cannot find an invariant circle in \eqref{example2} for
$\ep\ne0$ since the flux is $\ep$.
\begin{remark}
Note that if a map has a homotopically nontrivial invariant curve, then
the flux is zero (compute it for the curve). Conversely, if the flux is
zero, any homotopically non-trivial curve has
to have an intersection with its image.
(If it did not have any intersection, by Rolle's theorem, then the
image would always be in above or below the curve.)
The property that every curve intersects its image plays an important
role in KAM theory and is sometimes called {\sl intersection property}.
Besides area preserving and zero flux, there are other geometric assumptions
that imply the intersection property, notably, reversibility
of the map (see \cite{ArnoldS86})
\end{remark}
As a simple calculation shows, that perturbation in
\eqref{standardhamiltonian} is of the form $V' (q_n)$, with $V$
1-periodic --- therefore $\int_0^1 V' (q_n) \, dq_n = V(1) - V(0) = 0$
--- the flux of \eqref{standardhamiltonian}
is zero.
We see that even the possibility that there exist these quasi-periodic
orbits filling an invariant circle depends on geometric invariants.
Indeed, when we consider higher dimensional mechanical systems,
the analogue of area preservation is the preservation of a symplectic form,
the analogue of the flux is the Calabi invariant \cite{Calabi}
and the systems with zero Calabi invariant are called exact.
We point out, however, that the relation of the geometry to KAM theory
is somewhat subtle. Even if the above considerations
show that some amount of geometry is necessary,
they by no means show what the geometric structure is,
and much less hint on how it is to be incorporated in the proof.
The first widely used and generally
applicable method to study numerically
quasi-periodic orbits seems to have been the method of Lindstedt.
(We follow in this exposition \cite{FalcoliniL92b}.)
The basic idea of Lindstedt's method is to consider a family of
quasiperiodic functions depending on the parameter $\ep$ and to impose that
it becomes a solution of our equations of motion.
The resulting equation is solved -- in the sense of
power series in $\ep$ -- by equating terms with same powers
of $\ep$ on both sides of the equation. We will see how to
apply this procedure to
\eqref{standardhamiltonian} or \eqref{standardlagrangian}.
In the Hamiltonian formulation \eqref{standardhamiltonian},
\eqref{standardmap}
we seek $K_\ep :\TT^1 \to \RR \times \TT^1$
in such a way that
\mar{linstedthamilton}
\begin{equation}\label{linstedthamilton}
T_\ep \circ K_\ep (\theta) = K_\ep (\theta +\omega) \ .
\end{equation}
We set:
\begin{equation}\label{TaylorK}
K_\ep (\theta) = \sum_{n=0}^\infty \ep^n K_n (\theta)
\end{equation}
and try to solve by matching powers of $\ep$
on both sides of \eqref{linstedthamilton},
(after expanding $T_\ep\circ K_\ep(\theta)$
as much as possible in $\ep$ using the Taylor's theorem).
\footnote{The notation is somewhat unfortunate since
$K_n$ could mean both the $n$ term in the Taylor expansion
and $K_\ep$ evaluated for $\ep = n$.
In the discussion that follows, $K_1$,$K_2$, etc. will always refer to
the Taylor expansion. Note that $K_0$ is the same in both meanings. }
That is,
\begin{eqnarray*}
T_\ep \circ K_\ep (\theta)
& = & T_0\circ K_0 + \ep [T_1 \circ K_0 + (DT_0 \circ K_0) K_1]\\
&&\quad + \ep^2 [T_2 \circ K_0 + (DT_0\circ K_0) K_2 \\
&&\qquad\qquad + (DT_1\circ K_0) K_1
+ \frac{1}{2}(D^2 T_0\circ K_0) K_1^{\otimes2}] + \ldots \ .
\end{eqnarray*}
In the Lagrangian formulation \eqref{standardlagrangian}
we seek $g_\ep:\RR \to \RR$ satisfying
$g_\ep(\theta+1) = g_\ep(\theta)+1$
--- or, equivalently, $g_\ep(\theta) = \theta+\ell_\ep (\theta)$
with $\ell_\ep (\theta+1)=\ell_\ep(\theta)$,
i.e., $\ell_\ep:\TT^1 \to \TT^1$ ---
in such a way that
\mar{linstedtlagrange}
\begin{equation}
\label{linstedtlagrange}
\ell_\ep (\theta+\omega) + \ell_\ep(\theta-\omega) - 2\ell_\ep (\theta)
= - \ep V' (\theta+\ell_\ep(\theta))\ .
\end{equation}
If we find solutions of \eqref{linstedtlagrange}, we can
ensure that some orbits $q_n$ solving \eqref{standardlagrangian}
can be written as
$$q_n = n\omega + \ell_\ep (n\omega)\ .$$
Note that the fact that, when we choose coordinates on the circle,
we can put the origin at any place, implies that
$K_\ep(\cdot+\sigma)$ is a solution of \eqref{linstedthamilton}
if $K_\ep$ is,
and that $\ell_\ep (\cdot +\sigma)+\sigma$ is a solution of
\eqref{linstedtlagrange} if $\ell_\ep$ is.
Hence, we can -- and will -- always assume that
\begin{equation}
\label{normalization}
\int_0^1 \ell_\ep (\theta) \,d\theta=0 \ .
\end{equation}
This assumption, will not interfere with existence questions, since it
can always be adjusted, but will ensure uniqueness.
If we now write
\footnote{The same remark about the unfortunate notation we made in
\eqref{TaylorK} also applies here.}
$$\ell_\ep(\theta) = \sum_{n=0}^\infty \ell_n (\theta) \ep^n$$
and start matching powers, we see that matching the zero order terms yields
\mar{zeroorder}
\begin{equation} \label{zeroorder}
\begin{array}{c}
L_\omega \ell_0 (\theta)
\equiv \ell_0 (\theta+\omega) + \ell_0 (\theta-\omega)
- 2\ell_0 (\theta) = 0 \ , \\
\noalign{\vskip6pt}
\ds \int_0^1 \ell_0 (\theta) \,d\theta =0 \ .
\end{array}
\end{equation}
The operator $L_\omega$ n
\eqref{zeroorder}, which will appear repeatedly
in KAM theory, can be conveniently analyzed by using Fourier
coefficients. Note that
$$
L_\omega e^{2\pi ik\theta} = 2(\cos 2\pi k\omega-1) \,
e^{2\pi ik\theta} \ .
$$
Hence, if $\eta (\theta) = \sum_k \hat\eta_k e^{2\pi ik\theta}$,
then the equation
$$
L_\omega \varphi (\theta) = \eta (\theta)
$$
reduces formally to
$$
2 (\cos 2\pi k\omega-1) \, \hat \varphi_k = \hat\eta_k \ .
$$
We see that if $\omega \notin \que$, the equation \eqref{zeroorder} can be
solved formally in Fourier coefficients and $\ell_0 =0$.
(Later we will develop an analytic theory and describe precisely
conditions under which these solutions can indeed be
interpreted as functions.)
When $\omega \notin\que$, we see that $\cos 2\pi k\omega \ne1$
except when $k=0$.
Hence, even to write a solution we need $\hat\eta_0 = 0$,
and then we can write the formal solutions as
\mar{solution}
\begin{equation}\label{solution}
\hat\varphi_k = \frac{\hat \eta_k}{2(\cos 2\pi k\omega -1)} \ ,
\qquad k\ne0
\end{equation}
Note, however, that the status of the solution
\eqref{solution} is somewhat complicated
since $2\pi k\omega$ is dense on the circle and, hence, the denominator
in \eqref{solution} becomes arbitrarily small.
Nevertheless, provided that $\eta$ is a
trigonometric polynomial,
(See Exercise~\ref{trigpol} , where this is established under certain
circumstances)
and $\omega$ is irrational, we can solve
the equation \eqref{zeroorder}. In case that the \RHS is analytic
and that the number $\omega$ satisfies certain number
theoretic properties,
in Exercise~\ref{secondorder}, we can show that the solution is
analytic.
The equation obtained by matching $\ep^1$ is:
\mar{firstorder}
\begin{equation}\label{firstorder}
L_\omega\ell_1 (\theta) = - V' (\theta) \ ; \qquad
\int_0^1 \ell_1 (\theta)\,d\theta=0 \ .
\end{equation}
Since $\int_0^1 V' (\theta)\,d\theta =0$, we see that \eqref{firstorder}
admits a formal solution.
(Again, we note that the fact that $\int_0^1 V' (\theta)\,d\theta=0$
has a geometric interpretation as zero flux.)
Matching the $\ep^2$ terms, we obtain
\mar{secondorder}
\begin{equation}\label{secondorder}
L_\omega \ell_2 (\theta) = -V''(\theta) \ell_1 (\theta) \ ; \qquad
\int_0^1 \ell_2 (\theta) \,d\theta=0 \ ,
\end{equation}
and, more generally,
\mar{norder}
\begin{equation}\label{norder}
L_\omega \ell_n (\theta) = S_n (\theta) \ ; \qquad
\int_0^1 \ell_n (\theta) \, d\theta=0 \ ,
\end{equation}
where $S_n$ is an expression which involves derivatives of $V$ and terms
previously computed.
It is true (but by no means obvious) that
\begin{equation} \label{cancellation}
\int_0^1 S_n (\theta)\,d\theta=0,
\end{equation}
so that we can solve \eqref{norder} and proceed to
compute the series to all orders (when $\omega$ is
irrational and $S$ is a trigonometric polynomial or
when $\omega$ is Diophantine (see later)
and $S$ is analytic).
The fact that \eqref{cancellation} holds was already pointed out in
Vol II of \cite{Poincare}.
We will establish (\ref{cancellation})
directly by a seemingly miraculous
calculation, whose meaning will become clear
when we study the geometry of the problem.
(We hope that going through the messy calculation
first will give an appreciation for the geometric methods.
Similar calculations will appear in Section~\ref{Lagrangianmethod}.)
The desired result \eqref{cancellation} follows
if we realize that denoting
$\ell_\ep^{[\le n]}(\theta) = \sum_{i\le n} \ep^i \ell_i(\theta)$,
we have:
\begin{equation}\label{nordersum}
L_\omega \ell_\ep^{[\le n]} = \ep^n S_n
\end{equation}
Hence, multiplying \eqref{nordersum} by
$\left[1+\ell_\ep^{[\le n]}{}'(\theta)\right]$
and integrating, we obtain
\begin{equation}\label{totalintegral}
\begin{split}
0 &= \int_0^1 L_\omega \ell_\ep^{[\le n]}(\theta) \,d\theta
+ \int_0^1 L_\omega \ell_\ep^{[\le n]}(\theta) \,
\ell_\ep^{[\le n]}{}'(\theta)\,d\theta \\
&+ \int_0^1 V' (\theta + \ell_\ep^{[\le n]}(\theta))
\left[1 + \ell_\ep^{[\le n]}{}' (\theta) \right] \,d\theta \\
&-\ep^n \int_0^1 S_n (\theta) \,
\ell_\ep^{[\le n]}{}' (\theta) \,d\theta \\
&- \ep^n \int_0^1 S_n (\theta) \,d\theta \\
&+ O(\ep^{n+1}) \ .
\end{split}
\end{equation}
Now, we are going to use different arguments to show
that all the terms in \eqref{totalintegral}
except $\int S_n(\theta) \, d\theta$ vanish.
This will establish the desired result.
By changing variables in the integral we have:
\begin{equation}\label{integralV}
\int_0^1 V'(\theta + \ell_\ep^{[\le n]}(\theta))
\left[1+\ell^{[\le n]}_\ep{}'(\theta)\right] \,d\theta = 0.
\end{equation}
Furthermore, it is clear that
$\int_0^1 L_\omega \ell_\ep^{[\le n]}(\theta)\,d\theta=0$
because for any periodic function $f$
$\int_0^1 f(\theta)\,d\theta =
\int_0^1 f(\theta +\omega)\,d\theta =
\int_0^1 f(\theta -\omega)\,d\theta$
Noting that
$$
\int_0^1 \ell_\ep^{[\le n]}(\theta) \, \ell_\ep^{[\le n]}{}'(\theta)
= \int_0^1 \frac12
\biggl(\left[\ell_\ep^{[\le n]}(\theta)\right]^2\biggr)^{'}\,d\theta = 0
$$
and that
\begin{eqnarray*}
\int_0^1 \ell_\ep^{[\le n]} (\theta+\omega) \,
\ell_\ep^{[\le n]}{}'(\theta)\,d\theta
&=& -\int_0^1 \ell_\ep^{[\le n]}{}'(\theta+\omega) \,
\ell_\ep^{[\le n]}(\theta)\,d\theta\\
&=& - \int_0^1 \ell_\ep^{[\le n]}{}'(\theta) \,
\ell_\ep^{[\le n]}(\theta-\omega)\,d\theta \ ,
\end{eqnarray*}
we obtain that
$$
\int_0^1 L_\omega \ell_\ep^{[\le n]}(\theta) \,
\ell_\ep^{[\le n]}{}'(\theta)\,d\theta = 0.
$$
It is also clear that, because $\ell_0$ is a constant,
\begin{equation}\label{ishighorder}
\ep^n S_n(\theta) \ell_\ep^{[\le n]}{}' (\theta) = O(\ep^{n+1})
\end{equation}
Hence,
putting together \eqref{totalintegral}
and the subsequent identities, we obtain the desired conclusion that
$\int_0^1 S_n(\theta)\,d\theta$ vanishes.
\qed
\begin{remark}
There is a geometric interpretation for the vanishing of this integral.
One can compute the flux over the curve in the Hamiltonian formalism
predicted by $\ell_\ep^{[\le n]}(\theta)$. The fact that the flux vanishes is
equivalent to the fact that the integral vanishes.
\end{remark}
\begin{remark}
Note that it is rather remarkable that for every irrational frequency
we can find formal solutions (when the perturbation is a polynomial), or
for Diophantine frequencies for analytic perturbations.
Heuristically, this can be explained by the fact that, in area preserving
systems, we do not have small parts of the system controlling the long
term behavior (as it is the case in dissipative systems) and, hence,
perturbations still have to leave open many
possibilities for motion of the system.
When one applies the Lindstedt method to dissipative systems,
\cite{RandA87}, typically one sees that, except for a few
frequencies, the perturbation equations do not have a solution.
\end{remark}
\begin{remark}
The Lindstedt method can be used for dissipative systems
\cite{RandA87}. (Code for easy to use,
general purpose implementations is available from
\cite{RandAProg}.)
Then, one considers
$$
T_\ep \circ K_\ep (\theta) = K_\ep (\theta + \omega_\ep)\ .
$$
with $\omega_\ep = \sum \ep^n \omega_n$.
One has to choose the terms $\omega_0,\ldots,\omega_n$, so that
the equations \eqref{norder} have solutions.
It is a practical and easily implementable
method to compute limit cycles.
\end{remark}
\begin{exercise}\label{trigpol}
Show that if $V$ is a trigonometric polynomial,
then $l_n$ is also a trigonometric polynomial.
Moreover, $\deg(l_n) \le A n + B$ where $A$ and
$B$ are constants that depend only on the
degree of $V$. (For a trigonometric polynomial,
$V(\theta) = \sum_{|k| \le M} \hat V_k \exp(2\pi i k \theta)$,
the degree is $M$ when $\hat V_M \ne 0$ or
$\hat V_{-M} \ne 0.$)
As a consequence, if $V$ is a trigonometric polynomial
and $\omega$ is irrational, then the Lindstedt procedure
can be carried out to all orders.
\end{exercise}
\begin{remark}
The above procedure can be
carried out even in the case that
the function $V(x)$ is $e^{2\pi i x}$.
In this case, we obtain the so-called semi-standard map.
It can be easily shown that the trigonometric polynomials
that appear in the series only contain terms with
positive frequencies.
This makes the terms in the Lindstedt series
easier to analyze than those of the
case $V(x) = e^{2\pi i x} + e^{ - 2\pi i x}$.
Indeed, the analytical properties
of the term of the series
for $V(x) = e^{2\pi i x}$
very similar to those of
the normalization problem for a
polynomial.
We refer to
\cite{GreeneP81} for numerical
explorations, to \cite{Davie94} for rigorous upper
bounds of the radius of convergence and to
\cite{BerrettiM95},
\cite{BerrettiG98} for a method to transfer results from this
complex case to the real one.
\end{remark}
The convergence of the expansions obtained remains
at this stage of the argument we
have presented highly problematic.
Note that, at every stage, \eqref{norder} involves small divisors.
Worse still, the $S_n$'s are formed by multiplying terms obtained
through solving small divisor equations. Hence, the $S_n$
could be much bigger than the individual terms.
Poincar\'e undertook in
\cite{Poincare}, Paragraph 148
a study of the convergence of these series.
He obtained negative results for uniform convergence in a parameter
that also forced the frequency to change.
His conclusions read
(I transcribe the French as an example of the extremely nuanced
way in which Poincar\'e formulated the result.)
Roughly, he says that one can conclude that the series
does not converge, then points out that this has not been
proved rigorously and that there are cases that could be
left open, including quadratic irrationals.
The conclusion is that, even if the divergence has not
been proved, it is quite improbable.
\begin{quote}
Il semble donc permis de conclure que les series (2)
ne convergent pas.
Toutefois le raisonement qui pr\'ec\`ede ne suffit pas
pour \'etablir ce point avec une rigueur compl\'ete.
En effect, ce que nous avons d\'emontr\'e au ${\rm n}^{ {}_o}$ 42
c'est qu'il ne peut pas arriver que, pour toutes les
valeurs de $\mu$ inferieurs a une certaine limite, il y ait une
double infinit\'e de solutions p\'eriodiques, et il nous suffirait
ici que cette double infinit\'e exist\^ait pour une valeur de
$\hat \mu$ determin\'ee, different de $0$ et
g\'en\'eralment tr\'es petite.
[....]
Ne peut-il pas arriver que les series (2) converg\'ent
quand on donne aux $x^0_i$ certaines valeurs convenablement choisies?
Supposons, pour simplifier, qu'il y ait deux degrees de libert\'e;
les series ne pourraient-elles pas, par example, converger
quand $x^0_1$ et $x^0_2$ ont \'et\'e choisis de telle sorte que
le rapport $\frac{n_1}{n_2}$ soit incommensurable, et que son carr\'e
soit au contraire commensurable.
(ou quand le rapport $\frac{n_1}{n_2}$
est assujetti \'a une autre condition analogue \`a celle
que je viens d'ennoncer un peu au hassard)?
Les raisonnements de ce Chapitre ne me permettent pas
d'affirmer que ce fait ne se pr\'esentera pas.
Tout ce qu'il m'est permis de dire, cest qu'il
es fort inversemblable.
\end{quote}
This was remarkably prescient since indeed the series do converge
for Diophantine numbers. In particular,
for algebraic irrationals
(see Section~\ref{Diophantine_properties}, Theorem~\ref{Liouville}).
It is not difficult to show that,
for Diophantine frequencies, these series satisfy
estimates that fall short of
showing analyticity
\begin{equation}\label{Gevrey}
\|\ell_n\|_\sigma \le (n!)^{\nu}
\end{equation}
where $\nu$ is a positive number.
These estimates are sometimes called Gevrey estimates
and they appear very frequently in asymptotic analysis.
It is not difficult to construct examples
(indeed we present one in Exercise \ref{withoutgroup})
which have a similar structure and that the linearized
equation that we have to solve at each step satisfy
similar estimates. Nevertheless
they saturate \eqref{Gevrey}. Indeed, in many
apparently similar problems with a very similar
structure
(e.g. Birkhoff normal forms near a
fixed point, normal
forms near a torus, jets of center manifolds)
the bounds
\eqref{Gevrey} are saturated. We will not have
time to discuss these problems in these notes.
The proof of convergence of Lindstedt series was obtained in
\cite{Moser67} in a somewhat indirect way. Using the KAM
theory, it is shown that the solutions produced
by the KAM theory are analytic on the perturbation parameter.
It follows that the coefficients of the expansion have
to be the terms of the Lindstedt series and, therefore,
that the Lindstedt series are convergent.
The example in Exercise \ref{withoutgroup} shows that the
convergence that one finds in
KAM theory has to depend on the existence of
massive cancellations.
The direct study of the Lindstedt series was tackled successfully
in \cite{Eliasson96}. One needs to exhibit remarkable cancellations.
The papers \cite{Gallavotti94b} and \cite{ChierchiaF94}
contain another version of the cancellations above relating it
to methods of quantum field theory.
We note that the transformations that
reduce a map to its normal
Birkhoff normal form either near
a fixed point or near a torus were known
to diverge for a long time.
(See \cite{Siegel54}, \cite{Moser60}.)
Examples of divergence of asymptotic series
were constructed in \cite{Poincare}. To justify their
empirically observed usefulness, the same reference
developed a theory of asymptotic series, which
has a great importance even today.
It should be remarked that, at the moment of this
writing,
the convergence of Lindstedt series in
slightly different situations (lower dimensional tori
\cite{JorbaLZ00} or
the jets for center manifolds of positive definite systems
\cite{Mielke91}, p. 39)
are still open problems.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Siegel disks} \label{Siegel}
The following example is interesting because the geometry is reduced to a
minimum and only the analytical difficulties remain.
Not surprisingly, it was the first small divisors
problem to be solved \cite{Siegel42}, albeit with a
technique very different from KAM.
(Even if we will not discuss the original Siegel
technique in these notes, we point out that,
besides the original paper, there are
more modern expositions and extensions.
\cite{Brjuno71},\cite{Poschel86}.)
This problem is quite paradigmatic both for
KAM theory and for the theory of holomorphic dynamics.
In these lectures, we will discuss only the KAM
aspects and not the holomorphic dynamics.
A very good introduction to the problems
connected with Siegel theorem is
including both the KAM
aspects and the holomorphic dynamics
aspect is \cite{Herman87}. More up to date references are
\cite{Perezmarco92},
\cite{Yoccoz95}. The lectures \cite{Marmi99}
contain a great deal of material on the Siegel problem.
We consider analytic maps $f:\cee \to \cee$, $f(z) = az+N(z)$ with
$N(0) = 0$, $N'(0) = 0$,
and we are interested in studying their dynamics near the origin.
\medskip
When $|a|\ne1$, it is easy to show that the dynamics, up to an
analytic change of variables is that of $az$.
More precisely, there exists an $h: U \subset \cee\to\cee$, $h(0)=0$, $h'(0)=1$ and
\mar{semiconjugacy}
\begin{equation}\label{semiconjugacy}
f\circ h=h(az)
\end{equation} in a neighborhood of the origin.
The proof for $|a|>1$ can be easily obtained as follows (the case
$ 0 < |a|<1$ case follows by considering $f^{-1}$ in place of $f$).
We seek a fixed point of $h \mapsto f\circ h\circ a^{-1}$
on a space of functions
$h(z) = z+\Delta (z)$ with $\Delta (z)=O(z^2)$.
That is, we seek fixed points of the operator
$$
\Tau (\Delta) = a\, \Delta \circ a^{-1}
+ N\circ (\Id +\Delta)\circ a^{-1}\ .
$$
We note that, on a space of functions with
$\|\Delta\|_r = \sup_{|z|\le r} |\Delta (z)/z^2|$,
the operator $\Tau$ is a contraction if $r$ is sufficiently small.
Note that then $\Tau^r(0)$ converges uniformly on a ball
and the limit is analytic.
\begin{remark}
Note that the previous argument works without any significant change when
$f:\cee^d \to \cee^d$ and $a$ is a matrix all eigenvalues of which
have modulus less than 1. Indeed, a very
similar result for flows already appears
in Poincar\'e's thesis \cite{Poincare78}, where it was
established using the majorant method. (Remember that the concept of Banach
spaces had not been yet formalized, so that
fixed point proofs were unthinkable).
The method in \cite{Poincare78} can be adapted
without too much difficulty to cover the theorem
started above. Hence, the situation
when all the eigenvalues are smaller than one is sometimes
called {\em the Poincar\'e domain}.
\end{remark}
The situation that remains to be settled is that when $|a|=1$.
\begin{remark}
Building up on case for $|a| < 1 $, there is
a lovely proof by Yoccoz \cite{Herman87} using complex function
theory that one can extend the conjugacies for $|a| < 1$ to
a positive measure set
with $|a| = 1$. Several elements of this proof
can be used to obtain a very fast algorithm to
compute the so called {\sl Siegel radius}.
(See the definition in Proposition~\ref{univalent}.)
Another cute proof of a particular case of Siegel's theorem is in
\cite{Llave83} adapting a method of \cite{Herman86}. This method
can be applied to a variety of one-dimensional problems.
The method of \cite{Siegel42} has been quite refined
and extended in \cite{Brjuno71}, \cite{Brjuno72}.
We will not discuss the above proofs here,
because, in contrast with KAM ideas that have a
wide range of applications, they seem to be
rather restricted.
\end{remark}
It is typical of complex dynamics that there are very few possibilities
for the dynamics.
Either it is very unstable or it is a rigid rotation (up to a change of
variables).
We will prove something more general.
\mar{rigid}
\begin{lemma}\label{rigid}
Let $f:\cee^d\to\cee^d$ be analytic in a neighborhood of the origin and
$$
f(0)=0 \ , \qquad Df(0)=A \ ,
$$
where $A$ is a diagonal matrix with all the diagonal elements
of unit modulus (hence $\|A^{-n}\| = 1 \ \forall n \in \integer$).
Assume that there is a domain $U$, $0\in U$ and
a constant $K > 0$ such that
for all $n \in \nat$
\mar{equibounded1}
\begin{equation}\label{equibounded1}
\sup_{z\in U} |f^n(z)| \le K\ .
\end{equation}
Then there exists an analytic function $h:U\to \cee^d$ such that $h(0)=0$,
\begin{equation}\label{conclusion}
h'(0)=\Id\quad\quad, h\circ f=A\circ h.
\end{equation}
\end{lemma}
Of course, by the implicit function theorem
a solution of \eqref{conclusion}
implies that there is a solution of
\eqref{conjugacys} ($h$ in \eqref{conjugacys}
is the inverse of $h$ in \eqref{conclusion}).
Note also that the assumption \eqref{equibounded1} implies,
by Cauchy estimates that $|Df^n(0)| \le K'$, hence,
that all the eigenvalues are inside the closed unit
circle and that the eigenvalues on the unit circle
have trivial Jordan blocks. If rather than assuming
\eqref{equibounded1} for $n \in \nat$, we assumed
it for $n \in \integer$, this would imply the assumption
that $A$ is diagonal and has the eigenvalues on
the unit circle.
\begin{proof}
Consider
$$
h^{(n)} (z) = \frac{1}{n} \sum_{i=0}^n A^{-n} f^n(z)\ .
$$
Note that, using the definition of $A$ and
\eqref{equibounded1} and
\mar{normalization2} we have:
\mar{equibounded}
\mar{intertwine}
\begin{eqnarray}
&&h^{(n)} (0) =0\ ,\quad h^{(n)}{}' (0) =1\ , \label{normalization2}\\
&&\sup_{z\in U} |h^{(n)} (z) |\le K\ , \label{equibounded} \\
&&h^{(n)}\circ f(z) = Ah^{(n)} (z) + (1/n) [A^{-(n+1)} f^{n+1} (z)-z]\ .
\label{intertwine}
\end{eqnarray}
By \eqref{equibounded}, $h^{(n)}$ restricted to $U$ is a normal family and we
can find a subsequence converging uniformly on compact sets to a function
$h$.
Using \eqref{normalization2}, we obtain that $\tilde h(0)=0$, $\tilde h' (0)=1$.
Note also that,
since $\left| f^n (z)\right|$ is bounded independently of $n$,
by \eqref{equibounded1}
and so is $z$ for $z \in U$, we have that
$$
\frac{1}{n} [A^{-(n+1)} f^{n+1} (z)-z]
$$
converges to zero uniformly on any compact set
contained in $U$ as $n\to\infty$.
Therefore,
taking the limit $n\to\infty$ of \eqref{intertwine},
we obtain $h\circ f=A\circ h$.
\end{proof}
\begin{exercise}
Show that one can always assume that $U$ is
to be simply connected.
(Somewhat imprecisely, but
pictorially, if we are given are given a $U$ with holes,
we can always consider $\tilde U$ obtained by filling the holes
of $U$. The maximum modulus principle shows that $f^n$ is
uniformly bounded in $\tilde U$.)
In one dimension, show that the Riemann mapping that
sends $U$ into the unit disk and $0$ to itself
should satisfy \eqref{conclusion} except the normalization of
the derivative.
\end{exercise}
\begin{proposition}\label{uniqueness}
If the product of eigenvalues of $A$ is not another
eigenvalue, then
the function $\tilde h$ satisfying
\eqref{conclusion} is unique even in the sense of formal power series.
\end{proposition}
Note that, when $d = 1$ the condition of
Proposition \ref{uniqueness} reduces to the fact that
$A$ is not a root of unity. In particular, it is satisfied
when the modulus of $A$ is not equal to one.
When the modulus equals to $1$, the hypothesis of
Proposition \ref{uniqueness} reduces to $a$ not being
a root of unity, which is the same as
$a = \exp( 2 \pi i \theta) $ with $\theta \in \real -\rational$.
\begin{proof}
If we expand using the standard Taylor formula
for multi-variable functions,
$$
f(z) = \sum_{n=0}^\infty f_nz^{\otimes n}
$$
(where $f_n$ is a symmetric $n$-linear form taking values in $\cee^d$)
and seek a similar expansion for $\tilde h$, we notice that
$$
A\tilde h_n - \tilde h_n A^{\otimes n} = S_n \ ,
$$
where $S_n$ is a polynomial expression involving only the coefficients
of $f$ and $\tilde h_1 = \Id,\ldots,\tilde h_{n-1}$.
As it turns out, the spectrum of the operator $\L_A$
acting on $n$-multilinear forms by
\begin{equation} \label{action}
\tilde h_n \mapsto A \tilde h_n - \tilde h_n A^{\otimes n}
\end{equation}
is:
\begin{equation}\label{spectrumis}
\Spec(\L_A) =
a_i - a_{\sigma_1} \ldots a_{\sigma_n}, \quad
i\in \{ 1,\ldots,d \}, \quad
\sigma_1,\ldots,\sigma_n\in \{1,\ldots,d\},
\end{equation}
where $a_i$ denotes the eigenvalues of $A$.
See, e.g.,~\cite{Nelson69} for a detailed computation
which also leads to interesting algorithms.
We just indicate that the result can be obtained very
easily when the matrix is diagonalizable since one
can construct a complete set of eigenvalues of
\eqref{action} by taking products of eigenvalues of
$A$. The set of diagonalizable
matrices is dense on the space of
matrices. Hence the desired
identity between the spectrum of
\eqref{action} and the set described in
\eqref{spectrumis} holds in a dense set of matrices.
We also note that the spectrum is continuous
with respect to the linear operator.
\end{proof}
When $d=1$ and $|a|=1$, as
we mentioned before, the condition
for Proposition \ref{uniqueness}
(usually referred to as {\sl non-resonance condition})
reduces to:
$$
a= e^{2\pi i\omega}\ ,\qquad \omega\in \RR -\que\ .
$$
We note that, even if $a(a^{n-1}-1)\ne0$, it can be arbitrarily close
to zero, because $e^{2\pi i\omega (n-1)}$ is dense in the unit circle.
Hence, we also have small divisors in the computation of the $\tilde h_n$'s.
We note that when $d>1$, we can have small divisors if there is
some $|a_i|>1$, $|a_j|<1$ even if they are real.
When all $|a_j| =1$, $a_j = e^{2\pi i\omega_j}$, the non-resonance
condition amounts to
\mar{nonresonanceanalytic}
\begin{equation}\label{nonresonanceanalytic}
\sum_j k_j \omega_j\ne\omega_i \ , \
\forall k_j \in \nat, \sum_j k_j \ge 2
\end{equation}
We now investigate a few of the analyticity properties of $h$.
Of course, the power series expansion converges in a disk (perhaps
of zero radius) but we could worry about whether it is possible
to perform analytic continuation and obtain $h$ defined on
a larger domain.
\mar{univalent}
\begin{proposition} \label{univalent}
If $f$ is entire, the maximal domain of definition
of $h$ is invariant under
$A$.
In particular, when $d = 1$, $|a| = 1$, $a^n \ne 1$, the domain of
convergence is a disk. (The radius of the disk of convergence
of the function $h$ such that $h'(0) = 1$ is called the
Siegel radius.)
Moreover, when $d = 1$, $|a| \le 1$, $a^n \ne 1$,
the function $h$ is univalent in the domain of convergence.
\end{proposition}
\begin{proof}
To prove the first point, we just observe that if $f$ is entire and $h$
is analytic in the neighborhood of a point $z_0$, we can use the
functional equation \eqref{semiconjugacy} to define the function $h$ in
a neighborhood of $A\, z_0$.
Hence, if $h$ was defined in domain $D$
and $z_0 \in D$ was connected to the origin by a path $\gamma \subset
D$, we see that $A z_0$ is connected to the origin by $a \gamma
\subset a D$. We conclude that it is defined in $A D \cup D$ and that
the analytical continuation is unique. If we consider the maximal
domain of definition $A D \cup D \subset D$. Hence $A D = D$.
The second statement follows by observing that the only domains invariant
under an irrational rotation are disks.
To prove univalence, we assume that
if $h(z_1) = h(z_2)$
and one of them -- say $z_2$ -- different from $0$.
We want to conclude that $z_1 = z_2$.
Using \eqref{semiconjugacy}, we obtain
$h( a z_1) = h(a z_2)$. Repeating the process,
$h( a^n z_1 ) = h(a^n z_2) $.
Hence, when
$z \in \{ a^n z_2\}$, we have
\mar{contradiction}
\begin{equation} \label{contradiction}
h(z) = h( z \alpha)
\end{equation}
with $\alpha = z_1/z_2$. Since the set
where \eqref{contradiction} holds has an accumulation point:
when $|a| < 1 $, it accumulates to $0$, when $|a| = 1$ since
it is an irrational rotation, the orbits are dense on circles), we
conclude that it holds all over the unit disk.
Taking derivatives at $z = 0$, using $h'(0) = 1$, we obtain
$ \alpha = 1$.
\end{proof}
\mar{highd}
\begin{exercise}\label{highd}
Show that the conclusions of
Proposition \ref{univalent}
remain true if we consider
$d > 1$ and $A$
a diagonalizable matrix
with all eigenvalues in the unit disc
and satisfying \eqref{nonresonanceanalytic}.
Namely
\item[i')] The domain of definition is a polydisk.
\item[ii')] The function is univalent in its domain of
definition.
\end{exercise}
\begin{exercise}
Once we know that the domain of the function $h$ in
\eqref{semiconjugacy} is
a disk, the question is to obtain estimates of the radius.
Lower bounds are obtained from KAM theory.
Obtain upper bounds also using the fact that
by the Bieberbach-De Branges theorem,
the Taylor coefficients of a univalent function
satisfy upper bounds that depend on the radius of the disk.
On the other hand, we know the coefficients explicitly.
Also obtain upper bounds when $f(z) = a z + z^2 $ using the area formula
for univalent functions
$\Area h( B_r(0) ) = \pi \sum_{i=1}^\infty |h_i|^2 r^{2 i -2}$
knowing that the
range of $h$ -- orbits that are bounded -- cannot include any
point outside of the disk of radius $2$ and that we
know the coefficients $h_k$.
This exercise is carried out in great detail in
\cite{Rana87}, which established upper and lower bounds of
the radius for rotation by the golden mean.
\end{exercise}
It turns out to be very easy to produce examples where the series diverges.
We will discuss what we think is
oldest one \cite{Cremer28} (reproduced in
\cite{Blanchard84}). Other examples of \cite{Cremer38} can be found in
\cite{SiegelM95} Chapter 25 in a more modern form.
A different line of argument appears in
\cite{Ilyashenko79}, using more
complex analysis. This argument has been recently extended considerably
\cite{Perezmarco00}.
Consider $f(z) = az+z^2$ with $a=e^{2\pi i\omega}$,
then its $n^{\rm th}$ iteration is
$$
f^n (z) = a^n z+ \cdots + z^{2^n}\ .
$$
If we seek fixed points of $f^n$, different from zero, they satisfy
$(a^n -1) +\cdots + z^{2^n-1} =0$.
The product of the $2^n-1$ roots of this equation is $a^n-1$.
Hence, there is at least one root with modulus
smaller or equal to
$|a^n-1|^{1/(2^n-1)}$.
It is possible to find numbers $\omega \in \RR-\que$ such that
$$
\liminf_{n\to\infty} \,[\dist (n\omega,\nat)]^{1/(2^n-1)} =0\ .
$$
Hence, the $f$ above has periodic orbits different from zero in any
neighborhood of the origin.
This is a contradiction with $f$ being conjugate to an irrational rotation
in any neighborhood of the origin.
This shows that the perturbation expansions may diverge if the rotations
are very well approximated by rational numbers.
For complex polynomials in one variable it has been shown
\cite{Yoccoz95}, (see also \cite{Perezmarco92})
that if $\omega$ does not satisfy the
Brjuno conditions \eqref{Brjunocondition} below, the series for
the quadratic polynomial diverges.
The Theorem \ref{Siegel1} which we will prove
later will establish that if the condition
is met, then the series for all the non-linearities converges.
We say that $\omega$ satisfies
a Brjuno condition when there exists an
$\Omega$ increasing and log convex (the later properties are
just for convenience and can always be adjusted ) such that
\mar{Brjunocondition}
\begin{eqnarray}
&&\Omega (n) \ge \sup_{k\le n} |a^k -1|^{-1} \nonumber\\
&&\sum_n \frac{\log \Omega (2^n)}{2^n} <\infty\ \iff
\sum_n \frac{\log \Omega (n)}{n^2} <\infty\
\label{Brjunocondition}
\end{eqnarray}
The equivalence of the two forms of the condition is very easy from
Cauchy test for the convergence of series.
An example of
functions $\Omega(n)$ satisfying \eqref{Brjunocondition} is:
\[
\Omega(n) =
\exp( A n/(\log(n) \log\log(n) \cdots [\log^k(n)]^{1+\epsilon}) )
\]
for large enough $n$,
where by $\log^k$ we denote the function $\log$ applied $k$ times.
Indeed, \cite{Yoccoz95} shows that if
$\omega$ fails to satisfy the condition \eqref{Brjunocondition}
then $f(z) = e^{2 \pi i \omega} z + z^2$ is not linearizable
in any neighborhood of the origin.
\begin{remark}
In \cite{Yoccoz95} one can find the result that if,
a function $f(z)$ with $f(0) = 0$ ,
$f'(0) = a$, with $|a| = 1$ is not
linearizable, near $0$, then, the
quadratic function
$a z + z^2$ is not linearizable.
See also \cite{Perezmarco00}.
\end{remark}
In the case of one dimensional variables,
one can use the powerful theory of
continued fractions to express the Brjuno condition in an equivalent
form.
If $\omega \in \RR - \que$ can be written
$ \omega = [a_0,a_1,a_2,\cdots,a_n,\cdots]$
with $a_i \in \nat^+$,
we call $[a_0,a_1,\cdots,a_n] = p_n/q_n$
the convergents.
Brjuno condition is equivalent to
\mar{brjuno2}
\begin{equation}\label{brjuno2}
\B(\omega) \equiv \sum_n (\log q_{n+1})/q_n \le \infty
\end{equation}
A very similar condition
\mar{Perez-Marco2}
\begin{equation}\label{Perez-Marco2}
\sum_n (\log \log q_{n+1})/q_n \le \infty
\end{equation}
has been found in \cite{Perezmarco91} \cite{Perezmarco93}
to be necessary and sufficient for the existence of the
Cremer's phenomenon of accumulation of periodic
orbits near the origin in the sense that if condition
\eqref{Perez-Marco2} is satisfied, then, all non-linearizable
functions
have a sequence of periodic orbits accumulating at the
origin. If condition \eqref{Perez-Marco2} is not satisfied,
there exists a non-linearizable germ with no periodic
orbits other than zero in a neighborhood of zero.
\begin{remark}
We note that the formula \eqref{brjuno2} has very interesting
covariance properties under modular transformations.
They have been used quite successfully in
\cite{MarmiMY97}.
Without entering in many details,
we point out that another
function very closely related to the one
we have defined
satisfies (setting $\tilde \B(x) = +\infty$ when $x \in \rational$)
\begin{equation}\nonumber
\begin{split}
\tilde \B(\omega) &= -\log(x) + x \tilde \B( 1/x) \quad x \in (0,1/2) \\
\tilde \B(\omega)(-x)& = \tilde \B(x) \quad x \in (-1/2,0) \\
\tilde \B(\omega)(x+1) & = \tilde \B(x)
\end{split}
\end{equation}
Similar invariance properties are true
for the sum appearing in \eqref{Perez-Marco2}.
Nevertheless, it does not seem to have been investigated
as extensively.
Unfortunately, this one dimensional
theory does not have analogues in higher dimensions.
Some preliminary numerical explorations for the higher
dimensional case were done in \cite{Tompaidis96}.
\end{remark}
\begin{remark}
There is a very similar theory of changes
of variables that reduce the problem to linear
-- or some canonical -- form
for differential equations.
Of course, these normalizations resemble
the normalizations of singularity theory
and are basic for many applied questions such
as {\sl bifurcation theory}.
Similarly, there is a theory of these questions
in the $C^\infty$ or $C^r$ categories
under assumptions, which typically include that there are no
eigenvalues of unit length. This theory
usually goes under the name of
Sternberg theory.
The reduction of maps and
differential equations to {\sl normal form}
by means of changes of variables can also
be done when the map is required to preserve a
symplectic -- or another geometric -- structure
and one requires that the change of variables
preserve the same structure.
We will not discuss much of these interesting theories.
For more information on many of these topics
we refer to \cite{Brjuno89}, \cite{Bibikov79}.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}
In this Section, we will collect some background in analysis,
number theory and (symplectic and volume preserving) geometry.
Experts will presumably be familiar with most of the material
and will only need to read this as it is referenced in the
following text.
Of course, this chapter is not a substitute for manuals in
geometry or on analysis.
I have found \cite{Thirring97},
\cite{AbrahamMarsden}, \cite{GuilleminP74}
useful for geometry
and \cite{Stein70} \cite{Katznelson76} useful for analysis. Many of the
techniques are discussed in other papers in KAM theory
which we will mention as we proceed.
Specially the papers \cite{Moser66a}, \cite{Moser66b}
contain an excellent background in many of the analytical
techniques.
In the previous discussion of Lindstedt series we saw that we had
to consider
repeatedly equations of the form
$$L_\omega\varphi =\eta.$$
(The formal solution was given in \eqref{solution}.)
In this Section, we will also study equations
$$ D_\omega\varphi = \eta, \ \ {\rm where \ } \ \
D_\omega = \sum_i \omega_i \frac{\partial}{\partial\theta_i} \ ,
$$
which also appears in KAM theory.
A first step towards obtaining proofs of the KAM theorem
is to devise a theory of these equations.
That is, find conditions in $\omega$ and $\eta$ so that the function
defined by \eqref{solution} has a precise meaning.
The guiding heuristic principles are very simple:
\begin{itemize}
\item[1)]
The smoother the function $\eta$, the faster its Fourier coefficients
$\hat\eta_k$ decay.
\item[2)]
Some numbers $\omega$ are such that the
denominators appearing in
the solution \eqref{solution} do not grow very fast with $k$.
\item[3)]
Hence, for the numbers alluded to in 2), we
will be able to
make sense of the formal solutions \eqref{solution}
when the function considered is smooth.
\end{itemize}
We devote Sections \ref{prelimanalysis},
\ref{Diophantine_properties}, \ref{linear_estimates}
to making precise the points above.
We will need to discuss number theoretic properties
(usually called Diophantine properties) that
quantify how small the denominators can be as a
function of $k$. We will also need to study
characterizations of regularity in terms of
Fourier coefficients.
Since the result in KAM theory depends on the geometric properties
of the map -- as illustrated in \eqref{example1} and
\eqref{example2} -- it is clear that we will need to understand
which geometric properties enter in the conclusions.
Moreover, many of the traditional proofs indeed use a geometric formalism.
Hence, we have devoted a Section \ref{Geometric_structures}
to collect the facts we will need from differential geometry.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mar{prelimanalysis}
\subsection{Preliminaries in analysis} \label{prelimanalysis}
In modern analysis, it is customary to measure the regularity of a function
by saying that it belongs to some space in a certain scale of spaces.
Some scales that are widely used on compact manifolds are:
$$
\begin{array}{rcl}
C^r &\equiv& \Big\{\eta \,\Big|\,
D^r\eta \mbox{ is continuous, } \|\eta \|_{C^r}
\equiv \max \, \big(\sup_x |\eta (x)|,\ldots,\\
&&\qquad\qquad \sup_x |D^r\eta (x)| \big) \Big\}\ ,\\
%
\noalign{\vskip6pt}
C^{r+\alpha}& \equiv& \bigg\{\eta \,\Big|\, \|\eta\|_{C^r}
\equiv \max \biggl(\sup_x |\eta (x)|,\ldots,\sup_x |D^r\eta (x)|,\\[4mm]
&&\qquad\qquad
\sup_{x\ne y}\ds \frac{|D^r\eta (x) -D^r\eta(y)|}{|x-y|^\alpha}\biggr)\bigg\}\ , \\
%
\noalign{\vskip6pt}
A_\delta &\equiv& \Big\{\eta \,\Big|\,
\eta\mbox{ analytic on } |\Im \theta|<\delta,
\mbox{ continuous on } |\Im \theta|\le\delta, \\
&&\qquad\qquad \|\eta\|_\delta \equiv
\sup_{|\Im\theta|\le\delta}|\eta(\theta)|\Big\}\ , \\
%
\noalign{\vskip6pt}
H^s &\equiv& \Big\{\eta \,\Big|\, \eta\in L^2,\
(-\Delta +1)^{s/2} \eta\in L^2,\
\|\eta\|_{H^s} \equiv \|(-\Delta +1)^{s/2}\eta\|_{L^2}\Big\}\ ,
\end{array}
$$
where
$r=0,1,2,\ldots$, $\alpha\in [0,1]$, $\delta\in \RR^+$, $s\in\RR$
and we have used $\Delta$ to denote the Laplacian
Note that this notation (even if in wide usage) has certain ugly points.
$C^{r+0}$ and $C^{r+1}$ are ambiguous and can be
considered according to the first or the second definition.
Indeed, $C^{r+0}$ consider according to
the two definitions agrees as a space
(that is, the functions in one are functions in the
other and the topologies are the same), but the norms differ
(they are equivalent). On the other hand, $C^{r+1}$ differs
when the we interpret it in the first or in the second
sense. To avoid that, we will use $C^{r + \Lip}$
to identify the second definition.
All the above scales of spaces have advantages and disadvantages.
Against $C^{r+\alpha}$ we note that,
even if for $r = 0$,
these are the H\"older spaces which can
be defined in great generality (e.g.
metric spaces), when $r \ge 1$, the definition needs to be
done in a differentiable system of coordinates. This is because, for $r \ge 1$,
$D^r\eta (x)$ and $D^r\eta (y)$ are
multilinear operators in $T_xM$ and $T_yM$,
so that the differences in the definition are comparing
operators in different spaces.
Even though the different choices of coordinates lead to equivalent norms,
some of the geometric considerations are somehow cumbersome.
Also the composition operator --- ubiquitous in KAM theory ---
has properties which are cumbersome to trace in $C^{r+\alpha}$.
For example, the mapping $x\to f(x+ \cdot)$ can be discontinuous in
$C^{r+\alpha}$ when $f$ is $C^{r+\alpha}$.
It is somewhat unfortunate that the notations $C^r$ ($r \in \nat$ )
and $C^{r +\alpha}$, ($r \in \nat, \alpha \in [0,1) \cup \{\Lip \}$
suggest that one can consider perhaps $C^s$ ($ s \in \real$)
which includes both.
If one proceeds in this way,
one obtains very bad properties for the scale of spaces.
In colorful words, ``the limit of the space $C^{k+\alpha}$
as $\alpha\to 0$ is not $C^{k}$''. More precisely,
several important inequalities such as interpolation
inequalities which relate the different norms in a scale
fail to hold. Many characterizations -- e.g. in terms of
approximations by analytic functions -- break down
for the case that $r$ is an integer.
A possible way of breaking up the unfortunate $C^r$ vs.\ $C^{r+\alpha}$
notation is to introduce the spaces called $\Lambda_r$ in \cite{Stein70},
or $\hat C^r$ in
\cite{Zehnder75}, \cite{Moser66a}, \cite{Moser66b}.
In general we define:
\begin{equation} \label{Lambdar}
\begin{split}
&\Lambda_0 = C^0\\
&\Lambda_1 = \Big\{ f\,\Big|\, \sup_{\substack{1>|h|>0\\ x\in\RR}}
\frac{|f(x+h)+f(x-h)-2f(x)|}{h}
\equiv \|f\|_{\Lambda_1}<\infty\Big\} \\
&\Lambda_r = \{f \mid Df\in\Lambda_{r-1}, \ \|f\|_{\Lambda_r}
= \max (\|f\|_{C^0},\|Df\|_{\Lambda_{r-1}})\Big\}\ \ r \in \nat \\
&\Lambda_{r+\alpha} = C^{r +\alpha} \qquad r+\alpha \not\in \nat
\end{split}
\end{equation}
Here $[r]$ means the integer part of $r$ and
$\{r\}$ means the fractional part of $r$.
There are many reasons why the $\Lambda_\alpha$
spaces are the natural scale of spaces to
consider when one is considering
a space that includes the usual $C^{r + \alpha}$.
For example, one can obtain very nice approximation theory, interpolation
inequalities, and generalize naturally to several variables.
Note that
$$
C^1 \varsubsetneq C^{0+\Lip}\varsubsetneq \Lambda_1\ .
$$
Again, we point out that it is not easy to define these spaces on
manifolds except through patches.
Choosing different patches leads to different norms.
Fortunately, all of them are equivalent and, hence
define the same topology in the spaces.
Note that $C^r$ norms can be defined naturally on any smooth
Riemannian manifold.
(The norm of derivatives can be defined since it is the norm of
multilinear operators in the tangent bundle.)
The main inconvenience of $C^r$ ($r$ is, by assumption,
an integer) is that the characterization
by Fourier series is rather cumbersome.
It is easy to show integrating by parts that
\begin{eqnarray*}
\hat\eta_k & \equiv & \int_0^1 \eta (\theta) e^{-2\pi ik\theta}\,d\theta\\
&=& (-2\pi i)^{-r} k^{-r} \int_0^1 D^r \eta (\theta)e^{-2\pi ik\theta}\,
d\theta\ .
\end{eqnarray*}
Hence, if $\eta \in C^r$, we have
\mar{inequality1}
\begin{equation}\label{inequality1}
\sup_k \bigl( |\hat\eta_k|\, |k|^r \bigr) \le C_r \|\eta\|_{C^r}\ .
\end{equation}
where $C_r$ is a constant that depends only on $r$.
In the other direction, we have for
any $\delta > 0$
\mar{inequality2}
\begin{equation}
\begin{split}
\label{inequality2}
\|\eta\|_{C^r}
& \le \tilde C_r \sum_k |\hat\eta_k|\, |k|^r
= \tilde C_r \sum_k \frac{1}{|k|^{1+\delta}} \,
|\hat\eta_k|\, |k|^{r+1 +\delta} \\
& \le \tilde C_r \left( \sum_{k} \frac{1}{|k|^{1+\delta}} \right) \,
\sup_k \bigl(|\hat\eta_k|\, |k|^{r+1+\delta}\bigr) \\
&\le \tilde{\tilde C}_{r,\delta}
\sup_k \bigl(|\hat\eta_k|\, |k|^{r+1+\delta}\bigr) \ .
\end{split}
\end{equation}
Both inequalities \eqref{inequality1}, \eqref{inequality2}
are essentially optimal in the following sense.
Inequality \eqref{inequality1} is saturated by trigonometric polynomials,
while
the usual square wave --- or iterated integrals of it --- shows that
it is impossible to reduce the exponent on the right hand side of
\eqref{inequality2} to $r+1$.
This discrepancy is worse
when we consider functions on $\TT^d$, $d>1$.
In that case, to obtain convergence of the series,
in \eqref{inequality2} one
needs to take $\delta > d$.
This shows that studying regularity in terms of just the size of
the coefficients will lead to less than optimal results.
\begin{exercise}
Show that given any sequence $a_n$ of positive numbers converging to
zero, the set of continuous functions $f$
with $\limsup_k |\hat f_k| / a_k = \infty $ is
residual in $C^0$.
\end{exercise}
The spaces of analytic functions $A_\delta$ are better behaved
in respect of characterizations of norms of the function
in terms of its Fourier coefficients.
Integrating over an appropriate contour, we have Cauchy inequality
\mar{inequality1analytic}
\begin{equation}\label{inequality1analytic}
|\hat\eta_k| \le e^{-2\pi \delta |k|} \|\eta \|_{A_\delta}\ .
\end{equation}
On the other hand,
\mar{inequality2analytic}
\begin{equation}\label{inequality2analytic}
\begin{array}{rcl}
\|\eta\|_{A_{\delta-\sigma}}
& \le & \ds{\sum_{k\in\zed} e^{2\pi (\delta-\sigma)|k|} |\hat\eta_k|}\\[5mm]
&\le & \ds{\biggl( \sum_{k\in\zed} e^{-2\pi\sigma|k|}\biggr) \sup_k
e^{2\pi \delta |k|} |\hat\eta_k|} \\[5mm]
&\le & \ds{C\sigma^{-1} \sup_k e^{2\pi \delta|k|} |\hat\eta_k|}
\end{array}
\end{equation}
Of course, for Sobolev spaces, the characterization in terms of Fourier
coefficients is extremely clean:
$$\|\eta\|_{H^s} = \biggl( \sum_{k\in\zed}
(|k|^2 +1)^s |\hat\eta_k|^2\biggr)^{1/2}\ .$$
Sobolev spaces have other advantages.
For example, they are very well suited for numerical work and they also
work nicely with partial differential operators.
Many of the tools that we used in
$\Lambda_\alpha$ spaces also carry through to Sobolev spaces.
For example, we have the interpolation inequalities:
\mar{Sobolev-interp}
\begin{equation} \label{Sobolev-interp}
\|u\|_{H^{j}} \le K \| u \|_{H^m}^{j/m} \|u\|_{H^0}^{1 - j/m} \ .
\end{equation}
This inequality is a particular case of the
following Nirenberg inequality
\mar{nirenberg}
\begin{equation}\label{nirenberg}
\| D^i u \|_{L^r(\RR^n)} \le
C \| u \|_{L^p(\RR^n)}^{1 - i/m} \cdot \|D^m u \|_{L^q(\RR^n)}^{i/m}\ ,
\end{equation}
where $1/r = ( 1 - i/m)(1/p) + (i/m)(1/q)$.
We refer to \cite{Adams75}, p.~79.
These interpolation inequalities both for $\Lambda_\alpha$
and for Sobolev spaces are part of the more general
``complex interpolation method" and the scales of
spaces are ``interpolation spaces". Even if this is
quite important for certain problems of analysis in these spaces, we will
not go into these matters here.
As we will see later, some of the abstract versions of KAM as an implicit
function theorem work perfectly well for Sobolev spaces.
I think it is mainly a historical anomaly that these spaces are not
used more frequently in the KAM theory of dynamical
systems. (Notable exceptions are
\cite{Herman86}, \cite{KatznelsonO89a}.) Of course, for the applications of
Nash-Moser theory to PDE's or geometric problems, Sobolev spaces are used
quite often.
One of the most useful tools in the study of $C^{r+\alpha}$ spaces
is that they can be
characterized by their approximation properties by analytic functions.
The following characterization of $\Lambda_r$ spaces (remember that they
agree with the H\"older spaces $C^{[r]+\{r\}}$ when $\{r\}\ne0$) comes
{from} \cite{Moser66a,Moser66b}
(see also \cite{Zehnder75}, Lemma~2.2).
\mar{characterization}
\begin{lemma}\label{characterization}
Let $h\in C^0(\TT^1)$.
Then $h\in\Lambda_r$ if and only if for some $\sigma >0$ we can find a
sequence $h_i\in A_{\sigma2^{-i}}$ such that
\begin{itemize}
\item[i)] $\|h_i-h_0\|_{C^0}\to 0$
\vspace{2mm}
\item[ii)] $\sup_{i\ge1} (2^{ir}\|h_i-h_{i-1}\|_{A_{\sigma 2^{i-1}}})<\infty$
\end{itemize}
\end{lemma}
Moreover, it is possible to arrange that the sup in ii) is
equivalent to $\|h\|_{\Lambda_r}$ if one chooses the $h_i$ appropriately.
If we denote the sup in ii) by $M$, we have that for $h\in\Lambda_r$ it is
possible to find a sequence
$h_i$ in such a way that $M\le C_{\sigma,r}\|h\|_{\Lambda_r}$.
Conversely, for any
sequence $h_i$ as above, we have $\|h\|_{\Lambda_r}
\le \tilde C_{\sigma,r}M$.
Given a function $h\in\Lambda_r$ there are canonical ways of producing
the desired $h_j$.
For example, in \cite{Stein70} and \cite{Krantz83}
is shown that one can use
convolution with the Poisson kernel to produce
the $h_j$.
In that case, the sup in ii) can be taken to define a norm equivalent
to $\NORM_{\Lambda_r}$.
Another important feature of the $\Lambda_\alpha$ spaces is
that they admit a very efficient approximation theory.
The first naive idea that occurs to one
when trying to approximate a function by a smoother one is just
to expand in Fourier series and to keep only a finite number of terms
corresponding to the harmonics of small degree.
Indeed, for some methods of proof of the KAM theorem
that emphasize geometry this is the method of choice.
(See Section \ref{Arnoldmethod}.)
Unfortunately, keeping only a
finite number of the low order Fourier terms
is a much less efficient method of approximation
(from the point of view of the number of derivatives
required)
than convolving with a
smooth kernel.
Recall that summing a Fourier series is just convolution with the
Dirichlet kernel,
\begin{eqnarray*}
\sum_{k=-N}^N \hat\eta_k e^{2\pi ik\theta}
& = & \int_0^1 \eta (\varphi)\D_N (\theta-\varphi)\,d\varphi
= (\eta*D_N)(\theta)\\
\D_N(\theta) & = & \frac{\sin(2N+1)\pi\theta}{\sin\pi\theta} \ ,
\end{eqnarray*}
which is large and oscillatory and hence generates more oscillations
upon convolution than smooth kernels.
Hence the method of choice of approximating functions
by smoother ones is to
choose an positive analytic function
$K: \real^d \rightarrow \real$
decaying at infinity rather fast and
with integral $1$ and define
$K_t(x) \equiv \frac{1}{t^d} K(x/t)$.
We define smoothing operators
$S_t$ by convoluting with the kernels $K_t$. That is:
$$
S_t \phi = K_t * \phi.
$$
The properties of these smoothing operators that are
useful in KAM theory are (we express them in terms
of the $\Lambda_r$ spaces introduced in
\eqref{Lambdar}):
\mar{smoothing}
\begin{equation}\label{smoothing}
\begin{array}{ll}
\mbox{i)}&\quad
\lim_{t\to\infty} \|S_t u-u\|_{\Lambda_0} = 0\ , \quad u\in\Lambda_0\\
\noalign{\vskip6pt}
\mbox{ii)}&\quad
\|S_t u\|_{\Lambda_\mu} \le t^{\mu-\lambda} C_{\lambda\mu} \|u\|_{\Lambda_\lambda}\ ,
\quad u\in\Lambda_\lambda,\ 0\le\lambda\le \mu\\
\noalign{\vskip6pt}
\mbox{iii)}&\quad
\|(S_t-1)u\|_{\Lambda_\lambda}
\le t^{-(\mu-\lambda)} C_{\lambda\mu} \|u\|_{\Lambda_\mu}\ ,
\quad x\in\Lambda_\mu\ , 0\le \lambda\le\mu
\end{array}
\end{equation}
We note that a slightly weaker version of these properties is:
\mar{smoothinganalytic}
\begin{equation}\label{smoothinganalytic}
\begin{array}{ll}
\mbox{ii$'$)}&\quad \|S_tu \|_{\Lambda_{t^{-1}}}
\le k(\ell) \|u\|_{\Lambda_\ell}\qquad
t\ge 0\\
\noalign{\vskip6pt}
\mbox{iii$'$)}&\quad \|(S_\tau-S_t) u\|_{\Lambda_{\tau^{-1}}}
\le t^{-\ell} k(\ell)\|u\|_{\Lambda_\ell}
\qquad u\in\Lambda_\ell\quad \tau \ge t\ge1
\end{array}
\end{equation}
Note that it is easy to show that ii) $\Rightarrow$ ii$'$),
iii) $\Rightarrow$ iii$'$).
In \cite{Zehnder75} operators $S_t$ satisfying \eqref{smoothing} are said to
constitute a $C^\infty$ smoothing and those satisfying i), ii$'$), iii$'$)
a $C^\omega$ smoothing.
There are other smoothing operators and other
scales of spaces that satisfies the same inequalities.
Indeed, in the most abstract version of KAM theory,
which we discuss in Section \ref{Implicitfunction}, one
can even abstract these properties and obtain a general
proof which also applies to many other situations.
\medskip
One important consequence of
the existence of smoothing operators is
the existence of interpolation inequalities
(see \cite{Zehnder75}).
Even if this inequality were proved directly long
time ago, and can be obtained by different methods,
it is interesting to note that they are a consequence of
the existence of smoothing operators. As we mentioned, this
happens in other situations and for other spaces than
$\Lambda_r$.
In the following, we denote
$\| u \|_{r} \equiv \| u \|_{\Lambda_r} $.
\begin{lemma}\label{lem:interpolation}
For any $0 \le \lambda \le \mu$,
$0 \le \alpha \le 1$,
denoting
$$
\nu = (1-\alpha)\lambda +\alpha\mu
$$
we have for any $u \in \Lambda_\mu$:
\mar{interpolation}
\begin{equation} \label{interpolation}
\|u\|_{\nu} \le
C_{\alpha,\lambda,\mu} \|u\|_\lambda^{1-\alpha}\,
\|u\|_\mu^{\alpha}
\end{equation}
\end{lemma}
\proof
We clearly have:
$$\|u\|_\nu \le \|S_tu \|_\nu + \|(\Id - S_t)u\|_\nu\ .$$
Applying $ii)$ of
\eqref{smoothing} to the first
term and $iii)$ to the second, we obtain:
$$\|u\|_\nu \le t^{\nu-\lambda} C_{\alpha\lambda,\mu}\|u\|_\lambda
+ t^{-(\mu-\nu)} C_{\alpha\mu,\nu} \|u\|_\mu$$
and we obtain \eqref{interpolation}
by optimizing the right hand side in $t$.
\qed
These inequalities are descendents of inequalities for derivatives
of functions which were proved, in different versions,
by Hadamard and Kolmogorov and others.
For $\Lambda_r$, $r\notin\nat$ and for $C^r$, $r\in\nat$, the proofs can
be done by elementary methods and extend even to functions defined
in Banach spaces \cite{LlaveO99}. For analytic spaces, these interpolation
inequalities are classical in complex analysis and
are a consequence of the fact that the $\log | f(z)| $ is
sub-harmonic when $f(z)$ is analytic
\cite{Rudin87}.
In KAM theory the interpolation inequalities
\eqref{interpolation} are useful because if we have a smooth norm
$(\NORM_\mu)$ blowing up and a not so smooth one $(\NORM_\lambda)$ going
to zero, we can still get that other norms smoother than $\lambda$
still converge.
All the above results about
$\Lambda_\alpha$ spaces
of functions on the real
line can be generalized
to spaces of functions on $\RR^n$.
Indeed, one of the nicest
things of these spaces is that the theory
for them can be reduced to the study of one
dimensional restrictions of the function.
We refer to \cite{Stein70, Krantz83} for more details.
For analytic spaces, the theory can be
also extended with minor modifications.
In KAM theory we often have to
consider functions defined in
$\torus^m \times \real^n$ (often $n = m = d$).
In such a case, it is very convenient to
use expansions which are Taylor expansions in the real
variables and Fourier expansions in the angle:
\begin{equation}
\label{FourierTaylor}
f(\theta, I) = \sum_{j \in \nat^n, k \in \integer^m}
f_{j,k} I^j \exp( 2 \pi i k \cdot \theta)
\end{equation}
For these functions, it is convenient to define norms
\begin{equation}\label{analyticnorm}
\| f\|_\sigma =
\sup_{|I | \le e^{2 \pi \sigma}, |\Im(\theta)| \le \sigma}
|f(\theta, I)|
\end{equation}
With this definition, we have the Cauchy bounds
\begin{equation}
\label{Cauchyn}
\begin{split}
|f_{j,k} | &\le \exp( -2 \pi \delta( |j| + |k|) ) \|f\|_\sigma\\
\left|\left|\frac{\partial^{|r| +|s|}}
{ \partial I^r \partial \theta^s} f \right|\right|_{\sigma - \delta}
& \le C_{r,s,n,m} \delta^{-|r| -|s|}
\| f\|_\sigma
\end{split}
\end{equation}
The proof of these inequalities is quite standard
in complex analysis and will not be given in detail here.
It suffices to express the derivatives as integrals
over an $n+m$ dimensional torus which is close to
the boundary of the domain in which
$f(\theta, I)$ is controlled by $\|f\|_\sigma$.
The only subtlety is that
for some $l \in \{ 1,\ldots, m\}$,
$k_l > 0 $ one needs to choose
the torus $\Im(\theta_l) = -\sigma$.
(Similarly for the case when $k_l < 0$
one needs to choose the torus
$\Im(\theta_l) = \sigma$.)
It is also obvious that, under these supremum norm
the spaces constitute a Banach algebra,
that is:
\begin{equation}\label{banachalgebra}
\| f g \|_\sigma \le \| f\|_\sigma \|g\|_\sigma \ .
\end{equation}
Therefore, if $\|f\|_\sigma<1$, then
$\| (1+f)^{-1} \|_\sigma \le (1-\|f\|_\sigma)^{-1}$.
\subsection{Regularity of functions defined in closed sets. The
Whitney extension theorem}\label{Whitney}
In KAM theory, we often have to study functions defined in
Cantor sets. In particular, sets with empty interior.
In this situation, the concept of
Whitney differentiability plays an important role.
A reasonable notion of smooth functions
in closed sets is that they are the restriction
of smooth functions in open sets that contain them.
This definition is somewhat unsatisfactory since
the extension is not unique.
In the paper \cite{Whitney34a}, one can find an
intrinsic characterization of smooth functions in
a closed set.
\begin{definition}
We say that a function $f$ is $C^k$ in the sense of
Whitney in a compact set $F \subset \real^d$
when for every point $x \in F$ we can find
polynomials $P_x$ of degree less that $k$
such that
\begin{equation} \label{whitneybounds}
\begin{split}
& f(x) = P_x(x) \quad x \in F \\
& | D^i P_x(y) - D^i P_x(x)| \le |x -y|^{r-i} \sigma(|x -y|) \quad x, y \in F
\end{split}
\end{equation}
where $\sigma$ is a function that tends to zero.
\end{definition}
It is clear that if a function is the restriction of
a $C^k$ function the Taylor polynomials will do.
The deep theorem of \cite{Whitney34a} is that the converse is
true. That is,
\begin{theorem}\label{thm:Whitney}
Let $F \subset \real^d$ be a compact set.
If for a function $f$ we can find
polynomials satisfying \eqref{whitneybounds} and
such that $f(x) = P_x(x)$
then the function $f$ can be extended to
an a $C^r$ function in $\real^d$.
\end{theorem}
Note that if a function is $C^r$ in $\real^d$, then
one can find polynomials satisfying
\eqref{whitneybounds} by taking just the Taylor expansions
of $f$.
Contrary with what happened with the ordinary
derivatives, the polynomials satisfying \eqref{whitneybounds}
may not be unique. (For example, if we take $F$ to be the
$x$-axis in $\real^2$, we can take polynomials with a
a very different behavior in the $y$ direction.)
There are other variants of the definitions in which
rather than using $D^i P_x$ one introduces another polynomial
$P^i_x$ which is then, required to satisfy compatibility
conditions with the other polynomials.
Another variant useful for KAM theory appears in
\cite{LlaveV00}. It roughly states that, for Cantor sets with
a certain geometric structure, one just needs to verify
\eqref{whitneybounds} for $i = 0$.
The assumption that $F$ is compact can be removed.
It suffices to require \eqref{whitneybounds} in each
compact subset of $F$, allowing $\sigma$ to depend
on the compact subset.
In \cite{Stein70} one can find a version of this
theorem in which the extensions can be implemented
via a linear extension operator.
(There is a different extension operator $\E_k$ for each $k$.)
In \cite{Stein70}, one can also find versions for $C^{k+ \alpha}$.
The $C^\infty$ version can be found in \cite{Whitney34b}.
Even if adapting Whitney's theorem from real valued function to
functions taking values in a Banach space is well known,
(e.g \cite{Federer69} p. 225 ff.)
I do not know how to prove a similar result when
$F$ lies on an infinite dimensional space.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mar{Diophantineproperties}
\subsection{Diophantine properties} \label{Diophantine_properties}
In this Section, we want to study the existence of vectors $\omega\in\RR^n$
so that we can obtain upper bounds of $[\dist(\omega\cdot k,\nat)]^{-1}$
and of
$|\omega\cdot k|^{-1}$ when $k\in\zed^n - \{0\}$.
These are the small divisors that appear respectively
in the solution of the equations
\eqref{diff}, \eqref{der},
which appear often in KAM theory.
When we are studying problems such as those in Section~\ref{Siegel},
we need only to consider $k\in\nat^n$.
When $n=1$ for \eqref{diff} (and for $n=2$ for \eqref{der})
one can get quite good results using classical tools of number theory,
notably continued fractions, which we will not review here, in spite of their
importance in 1-dimensional dynamics.
For example, the classical result of Liouville states
\mar{Liouville}
\begin{theorem}\label{Liouville}
Let $\omega\in \RR-\que$ satisfy $P(\omega) =0$ with $P$ a polynomial
of degree $\ell$ with integer coefficients.
Assume that $P'(\omega)=0,\ldots,P^{(j}(\omega)=0$, $P^{(j+1}(\omega)\ne0$.
Then for some $C>0$
\begin{equation}\label{Diophantinebounds}
\Big| \omega -\frac{m}n\Big| \ge Cn^{-\ell/(j+1)} \qquad \forall\ m,n\in\zed\ .
\end{equation}
\end{theorem}
\begin{proof}
The zeroes of polynomials are isolated, hence $P(\frac{m}{n})\ne0$
when $\frac{m}{n}$ is close enough to $\omega$.
This together with the fact that $n^\ell P(\frac{m}{n})\in\zed$
implies that $|n^\ell P(\frac{m}{n})|\ge1$
and, therefore, $|P(\frac{m}{n})-P(\omega)| \ge n^{-\ell}$.
On the other hand, by the Taylor's theorem,
$$
\Big|P\Big(\frac{m}n\Big)-P(\omega)\Big|
\le C\Big|\omega-\frac{m}n\Big|^{j+1}
$$
for some $C>0$.
(The \RHS is the remainder of Taylor's theorem.)
This yields the desired result for $\frac{m}{n}$ close to $\omega$.
For $\frac{m}{n}$ far from $\omega$, the result is obvious.
\end{proof}
Theorem~\ref{Liouville} was significantly improved by Roth,
who showed that, if $\omega$ is
an algebraic irrational,
$|\omega - \frac{m}n|\ge C_\ep n^{-2-\ep}$ for every $\ep > 0$.
The numbers that satisfy the equation \eqref{Diophantinebounds}
in the conclusions of
Theorem~\ref{Liouville} are quite important
in number theory and in KAM theory and are
called Diophantine. As we will see in Lemma~\ref{Diophantine},
Diophantine
numbers occupy positive measure, hence, there
are some of them which do not satisfy the
hypothesis of Theorem \ref{Liouville}.
\mar{def:Diophcond}
\begin{definition}\label{def:Diophcond}
A number $\omega$ is called
{\em Diophantine} of type $(K,\nu)$
for $K>0$ and $\nu\geq1$, if
\mar{diophantinebound}
\begin{equation}\label{diophantinebound}
\left| \omega - \frac pq \right| > K \, |q|^{-1-\nu}
\end{equation}
for all $\frac pq \in \que$.
We will denote by $\D_{K,\nu}$ the set of
numbers that satisfy
\eqref{diophantinebound}.
We denote by $\D_\nu = \cup_{K> 0} \D_{K,\nu}$.
A number which is not Diophantine
is called a {\em Liouville number}.
\end{definition}
The numbers $\omega$ for which
$|\omega-\frac{m}n|\ge Cn^{-2}$ are called ``constant type''
and the previous result shows that quadratic irrationals are constant type.
It is an open problem to decide whether $\root 3 \of 2$ is constant
type or not. Indeed, it would be quite interesting
to produce any non-quadratic algebraic number
which is of constant type.
In higher dimensions, there are two types of
Diophantine conditions that appear in
KAM theory, namely:
\begin{eqnarray}
&|\omega \cdot k |^{-1} \le C |k|^\nu \quad \forall k \in \integer^n - \{0\}
\label{diophantineflow} \\
&|\omega \cdot k - \ell |^{-1} \le C |k|^\nu \quad
\forall (k, \ell) \in \integer^n \times \integer - \{ (0,0) \}
\label{diophantinemap}
\end{eqnarray}
The first condition \eqref{diophantineflow}
appears when we consider the KAM theory
for flows, the second one \eqref{diophantinemap} when we consider KAM theory
for maps. As we will see the arguments are very similar in both
cases.
\begin{remark}
One important difference between these Diophantine
conditions is that the first
condition \eqref{diophantineflow} is maintained
-- with only different constants --
if the vector $\omega$ is multiplied by a constant.
Nevertheless, the second one is not. Indeed, if we
take advantage of this to set one of the coordinates to 1,
then, we see that \eqref{diophantineflow} becomes
\eqref{diophantinemap} for the vector in one dimension less
obtained by keeping the coordinates not set to $1$.
The arguments that study geometry of these Diophantine conditions
are identical.
Nevertheless, we point out that
the scale invariance of \eqref{diophantineflow}
will have some consequences later, namely
that KAM tori for flows often appear
in smooth one-dimensional families, whereas those for maps
are isolated.
\end{remark}
For us, the most important result is
\mar{Diophantine}
\begin{lemma}\label{Diophantine}
Let $\Omega :\RR\to\RR$ be an increasing function satisfying
\mar{summability}
\begin{equation}\label{summability}
\sum_{r=1}^\infty \Omega(r)^{-1} r^{n-1} n$ and
for $K$ sufficiently big.
This shows that the set of
Diophantine numbers
$\D_\nu$ has full measure for $\nu > n$.
Indeed
\mar{density}
\begin{equation}\label{density}
| \C \cap \D_{K,\nu} | \ge 1 - K b(\nu,n)
\end{equation}
\begin{proof}
We will denote by $\sigma_n$ constants that depend only on the
dimension~$n$. The same symbol can be used for
different constants.
For $k\in \zed^n\setminus\{0\}$, $\ell\in\zed$ we consider the set
$$
\B_{k,\ell} = \{\omega\in\RR^n \ | \ |\omega\cdot k-\ell|
\le \Omega (|k|)^{-1}\}
$$
consisting of the $\omega$'s for which the desired inequality
\eqref{denbound} fails precisely for $k,\ell$.
The desired set will be the intersection of the complements of these sets.
Geometrically $\B_{k,\ell}$ is a strip bounded by parallel planes which
are at a distance $2\Omega(|k|)^{-1}|k|^{-1}$ apart
(see Figure~\ref{parallel-fig}).
%
\mar{parallel-fig}
\begin{figure}
\centerline{\epsfysize=4.0truein\epsfbox{llave-fig2.eps}}
\label{parallel-fig}
\caption{}
\end{figure}
%
Thus, given a unit cube $\C\in\RR^n$, the measure of $\C\cap
\B_{k,\ell}$ cannot exceed $\sigma_n\Omega (|k|)^{-1}|k|^{-1}$.
We also observe that given $k\in \zed^n - \{0\}$, there is only a finite
number of $\ell$ such that $\C\cap \B_{k,\ell} \ne\emptyset$.
Indeed, this number can be bounded by $\sigma_n|k|$.
Therefore, for any $k\in\zed^n\setminus\{0\}$
$$
\sum_{\ell\in\zed} |\B_{k,\ell} \cap \C| \le \sigma_n \Omega(|k|)^{-1} \ ,
$$
hence,
\begin{equation}
\begin{split}
1 - |\C \cap \D_\Omega|
= & \sum_{k\in\zed^n} \sum_{\ell\in\zed\setminus\{0\}}
|\B_{k,\ell} \cap \C| \\
& \le \sigma_n \sum_{k\in\zed^n\setminus \{0\}} \Omega(|k|)^{-1} \\
& \le \sigma_n \sum_{r=1}^{\infty} \Omega(r)^{-1} r^{n-1} \ .
\end{split}
\end{equation}
Under the hypothesis that the \RHS of the above
equation is smaller than $1$, the conclusions hold.
\end{proof}
An important generalization of the above argument
\cite{Pjartli69} leads to the conclusion
that a submanifold of Euclidean space
that has curvature (or torsion or any other
higher order condition) in such a way that planes
cannot have a high order tangency to it
(see below or see the references)
then the submanifold has to contain Diophantine numbers.
Even if the proof is relatively simple, the abundance of
Diophantine numbers in lower dimensional curves has
very deep consequences since it allows one to
reduce the number of free parameters needed in
KAM proofs.
\begin{lemma}\label{pjartli}
Let $\Sigma$ be a compact $C^{l+1}$ submanifold of
$\real^n$.
Assume that at every point $x \in \Sigma$ of the manifold
\begin{equation}\label{torsion}
T_x\Sigma + T^2_x \Sigma + \cdots + T^l_x \Sigma = T_x \real^n
\end{equation}
where by $T^j_x \Sigma$
we denote the $j$ tangent plane to $\Sigma$.
Then, we can find a constant $C_\Sigma$ that depends only on
the manifold such that:
$$
| \Sigma - \D_\Omega| \le C_\Sigma \sum_{r=1}^\infty \Omega(r)^{1/l} r^{n -1}
$$
where by $| \cdot |$ we denote the Riemannian volume of the
manifold.
\end{lemma}
The geometric meaning of the hypothesis
\eqref{torsion} is that the manifold is
not too flat and that it has curvature
and torsion (or torsion of
high order) so that every neighborhood of a point has to
explore all the directions in space.
In particular, we will have a lower bound on the area of
the portion of the manifold that can be
trapped in a resonant region, which in the
space of $\omega$ is a flat plane.
The remaining details of the
proof is left as an exercise for the interested reader.
See also the lectures on number theory in this volume.
The proof follows by noting that because of \eqref{torsion}
we can bound the measure of the
regions $ \sum_{l \in \integer} \Sigma \cup \B_{k,l} \le C_\Sigma
\Omega(k)^{-1/l}$.
The worst case happens when the manifold is tangent to a very high order
to one of the resonant regions. Since the order of tangency
-- as well as the constants involved -- are uniformly bounded,
we obtain the desired result.
\begin{remark}
Notice that the formulation of
the Diophantine properties
\eqref{diophantineflow} and \eqref{diophantinemap}
also makes sense if we allow $\omega$ to take
complex values. This sometimes appears when
we study complex maps and it is a useful
tool. Notice that the argument we have presented
works very similarly for the case of $\omega$ taking
complex values. Indeed, the norm of the inverse
can be bounded by the norm of inverse of the real
part (or the norm of the inverse of the imaginary
part) so, when the real or imaginary parts
of an $\omega$ vector are Diophantine, the vector
is Diophantine.
Sometimes, when studying problems with polynomials
we will also need the inequalities
only for $k \in \nat^n$. Needless to say, these
are much easier to satisfy since the signs
have less possibilities to compensate and lead to
small numbers.
\end{remark}
\begin{exercise}
Construct a complex vector which is Diophantine,
but whose imaginary and real parts are not
Diophantine.
\end{exercise}
\begin{remark}
The same simple minded argument used in the proof of
Lemma \ref{Diophantine} can be used to obtain
estimates not only on the Lebesgue measure of
the set of Diophantine numbers but also other
geometric properties (for example Hausdorff measure),
of sets satisfying Diophantine properties,
and that are
forced to belong to a manifold, have a resonance,
etc.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Estimates for the linearized equation} \label{linear_estimates}
In this subsection, we will consider estimates for the
following equations \eqref{der}, \eqref{diff} that occur
very frequently in KAM theory.
We have encounter them already in the
study of Lindstedt series and we will encounter them
again as linearized equations.
We will consider equations of the form:
\mar{der}\mar{diff}
\begin{gather}
D_\omega \varphi =\eta\qquad \left( D\omega \equiv \omega_1
\frac{\partial}{\partial\theta_1} + \cdots
+ \omega_n\frac{\partial}{\partial\theta_n}\right) \label{der}\\[2mm]
\begin{array}{rcl}
L_\omega\varphi =\eta\quad&&\Bigl(L_\omega \varphi (\theta_1,\ldots,\theta_n)
\\[2mm]
&&\quad \equiv \varphi (\theta_1 +\omega_1,\ldots,\theta_n+\omega_n)
- \varphi (\theta_1,\ldots,\theta_n)\Bigr)\ ,
\end{array}\label{diff}
\end{gather}
where $\eta :\TT^n\to\RR$ is given and the unknown function
to be found is $\varphi$.
For the sake of simplicity we will only discuss in detail \eqref{der}.
The same considerations apply for \eqref{diff} and we will indicate
the minor differences -- in fact simplifications -- that
enter in the discussion of \eqref{diff}.
Recall that these equations have a formal solution in terms of Fourier series.
Namely, if
$$\eta (\theta) = \sum_{k\in\zed^n} \hat\eta_k
e^{2\pi i k\cdot\theta}\ ,
\qquad \hat\eta_0=0\ ,$$
then any reasonable solution of \eqref{der}
for which one can define unique Fourier coefficients
(e.g. any distribution) has
to satisfy:
$$
\hat\varphi_k 2\pi i k\cdot\omega = \hat\eta_k\ .
$$
Hence, if $k\cdot\omega\ne0$, then\mar{solution2}
\begin{equation}
\label{solution2}
\hat\varphi_k = \frac{\hat{\eta}_k}{2\pi i k\cdot\omega}\ .
\end{equation}
We restrict our attention to cases when $k\cdot\omega\ne0$ for any
$k\in\zed^n-\{0\}$.
In that case $\varphi$ is determined by \eqref{solution2} up to an additive
constant since we can take any $\hat\varphi_0$.
To avoid unnecessary complications, we will set $\hat\varphi_0=0$.
It is not difficult to see that, unless we impose some quantitative
restriction on how fast
$|k \cdot\omega|^{-1}$
can grow, the solutions given by (\ref{solution2}) may fail to be even
distributions.
E.g., take $\hat\eta_k = e^{-|k|}$ and arrange that there are
infinitely many $k$ for which
$|k \cdot\omega|^{-1} \ge e^{e^{|k|}}$.
\begin{exercise}
Given any sequence $a_n$ of positive terms tending to infinity
construct an $\omega \in \real^n - \rational^n $
such that, for infinitely many $k \in \integer^n $
\begin{equation}
|\omega \cdot k |^{-1} \ge a_{|k|}.
\end{equation}
Show that the $\omega$ constructed above are dense
(even if, as we have shown, they will be of measure zero for
sequences $a_n$ which grow fast enough).
\end{exercise}
We will consider $\omega$ which satisfy\mar{Diophantinecond}
\begin{equation}
\label{Diophantinecond}
|k \cdot\omega|^{-1} \le \gamma |k|^\nu \ .
\end{equation}
These numbers were studied in Section~\ref{Diophantine_properties}.
It is not difficult to obtain some crude bounds for analytic or finite
differentiable functions (we will do better later).
Recall that for $\eta\in A_\delta$
$$|\hat\eta_k| \le e^{-2\pi\delta|k|} \|\eta\|_{A_\delta}\ ,$$
while for $\eta\in C^r$
$$|\hat\eta_k| \le (2\pi)^{-r} |k|^{-r} \|\eta\|_{C^r}\ .$$
Hence, if $\omega$ satisfies \eqref{Diophantinecond}, we have for
$\eta\in A_\delta$
$$|\hat\varphi_k| \le (2\pi)^{-1}\gamma |k|^\nu
e^{-2\pi \delta|k|} \|\eta\|_{A_\delta}\ ,$$
and for $\eta\in C^r$
$$|\hat\varphi_k| \le (2\pi)^{-r-1}
\gamma |k|^{\nu-r}\|\eta\|_{C^r}\ .$$
These estimates do not allow us to conclude that $\varphi$ belongs to the
same space as $\eta$, but allow us to conclude that it belongs to a
slightly weaker space.
As mentioned before, the characterization of the analytic
spaces in terms of their Fourier series is very
clean, so that we can obtain
estimates of the solutions in these spaces.
Then, we will use
Lemma \ref{characterization}
to obtain the results for $\Lambda_r$ spaces.
Since for $0 < \sigma < \delta$ we have:
$$\|e^{2\pi i k\cdot\theta}\|_{\delta-\sigma}
\le e^{2\pi (\delta-\sigma)|k|}\ ,$$
we have\mar{analytic}
\begin{equation}
\label{analytic}
\begin{array}{rcl}
\|\varphi\|_{A_{\delta-\sigma}}
&\le &\ds \sum_{k\in\zed^n\setminus\{0\}} |\hat\varphi_k|
e^{2\pi |k| (\delta-\sigma)} \\
\noalign{\vskip6pt}
&\le &\ds \sum_{k\in\zed^n\setminus\{0\}}
\frac1{2\pi |k\cdot\omega|} \|\eta\|_{A_\delta}
e^{-2\pi \sigma|k|}\\
\noalign{\vskip6pt}
&\le &\ds \frac{1}{2\pi}\gamma\|\eta\|_{A_\delta} \sum_{k\in\zed^n\setminus\{0\}}
|k|^\nu e^{-2\pi \sigma|k|}\\
\noalign{\vskip6pt}
&\le &\ds C\gamma\|\eta\|_{A_\delta} \sum_{\ell\in\nat}
\ell^{\nu +n-1} e^{-2\pi \sigma\ell}\\
\noalign{\vskip6pt}
&\le &\ds C\gamma\sigma^{-(\nu+n)} \|\eta\|_{A_\delta}\ ,
\end{array}
\end{equation}
where in the fourth inequality we have just used that we do first the sum
in the $k$ with $|k|=\ell$ (the number of terms in this sum can
be bounded by $C\ell^{n-1}$).
We denote by $C$ constants that depend only on $\nu$ and the dimension $n$
and are independent of $\gamma,k$, etc.
Similarly, using that
$$\|e^{2\pi i k\cdot\theta}\|_{C^s} \le C|k|^s\ ,$$
we have
\footnote{Here, $C$ depends on $s$ even if it is independent of
$k$. We, however do not include the $s$
dependence in the notation to avoid clutter.}
\begin{eqnarray*}
\|\varphi\|_{C^s}
&\le & C\gamma \|\eta\|_{C^r} \sum_{k\in\zed^n} |k|^{\nu-r+s}\\
&\le & C\gamma \|\eta\|_{C^r} \sum_{\ell\in\nat}
\ell^{\nu -r+s+n-1}\ .
\end{eqnarray*}
The sum in the \RHS converges provided that
$$r>\nu +s +n\ .$$
Actually, one can do significantly better that these
crude bounds if one notices that
the small divisors have to appear rather infrequently
(see \cite{Russmann75,Russmann76}).
Note that $\omega \cdot(k + \ell) = \omega \cdot k + \omega \cdot \ell$.
Hence, if $\omega \cdot k$ happens to be very small,
$\omega \cdot(k + \ell) \approx \omega \cdot \ell$,
so that if $|\ell| << |k|$,
$\omega\cdot (k + \ell) \approx \omega \cdot \ell$.
In other words, the really bad small divisors appear
surrounded by a ball on which the divisors are not that
small. Hence, if instead of estimating the size
as in \eqref{analytic} using the estimates \eqref{diophantineflow}
in the third step we use a Cauchy-Schwartz inequality,
which takes into account the sum of
terms, not just the
the sup and that can profit from the
fact that \eqref{diophantineflow} cannot
be saturated very often, we obtain the result of
\cite{Russmann75,Russmann76}, which reads:
\begin{lemma}\label{linearestimates}
Assume that
$\omega$ satisfies \eqref{diophantineflow},
with $\nu \ge n-1$
and that
$\tilde \omega$ satisfies \eqref{diophantinemap}.
Let $\eta,\tilde \eta$ be analytic functions
with zero average.
Then, we can find $\varphi, \tilde \varphi$ solving
\eqref{der}, \eqref{diff}.
Namely
\begin{equation}
\begin{split}
&D_\omega \varphi = \eta \\
& L_{\tilde \omega} \tilde \varphi = \tilde \eta.
\end{split}
\end{equation}
and $\varphi$, $\tilde \varphi$ have zero average.
Moreover, we have for all $\delta > 0$:
\begin{equation} \label{analyticcohomology}
\begin{split}
&\| \varphi \|_{\sigma - \delta} \le C \delta^{-\nu} K_{\nu, n} \| \eta\|_\sigma \\
&\| \tilde \varphi \|_{\sigma - \delta} \le C \delta^{-\nu} K_{\nu, n} \|\tilde \eta\|_\sigma
\end{split}
\end{equation}
Where the $C$ are the same constants that appear in \eqref{diophantineflow},
\eqref{diophantinemap} and $K$ are constants that depend
(in a very explicit formula) only
on the exponent in \eqref{diophantineflow}, \eqref{diophantinemap}
and the dimension of the space.
If we assume that $\eta$, $\tilde \eta$ are in $\Lambda_r$,
$r > \nu$, we obtain:
\begin{equation} \label{differentiablecohomology}
\begin{split}
& \| \varphi\|_{\Lambda_{r -\nu}} \le
C K_{\nu,n} \|\eta\|_{\Lambda_r} \\
& \| \tilde \varphi\|_{\Lambda_{r -\nu}} \le
C K_{\nu,n} \| \tilde \eta\|_{\Lambda_r}
\end{split}
\end{equation}
\end{lemma}
We just note that the part \eqref{differentiablecohomology}
is a consequence of \eqref{analyticcohomology} using the
the characterization of
differentiable functions by properties of the approximation
by analytic functions in
Lemma \ref{characterization}.
When studying analytic problems, one can be sloppy with the
exponents obtained and still arrive at the same result.
However,
as \eqref{differentiablecohomology} shows, taking care of
the exponents is crucial if we are studying finitely differentiable
problems and want to obtain regularity which is close to optimal.
\begin{exercise}
Read the argument in \cite{Russmann76}. Do you obtain
some improvement using the H\"older inequality in place
of Cauchy-Schwartz?
\end{exercise}
\begin{exercise} \label{secondordereq}
In the study of Lindstedt series
(e.g. \eqref{norder})
we encountered second order
equations for $\varphi$
given $\eta$ of the form:
\begin{equation} \label{eq:secondordereq}
\varphi(x + \omega) + \varphi(x - \omega) - 2 \varphi(x) = \eta(x)
\end{equation}
where $\varphi$ and $\eta$ are periodic and
$\omega$ is a Diophantine number.
Develop a theory of the equation
\eqref{eq:secondordereq}
along the theory of the theory developed
in Lemma~\ref{linearestimates}.
Do it either by treating it directly in
Fourier series or by
factoring it as two
equations:
\begin{equation}
\begin{split}
& w(x) - w(x- \omega) = \eta(x) \\
& \varphi(x +\omega) - \varphi(x) = w(x)
\end{split}
\end{equation}
Are there any differences between the estimates or
the solvability conditions you get by the two methods?
What happens if instead of using the naive estimates
presented in the text you use the estimates of
\cite{Russman76}?
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mar{Geometricstructures}
\subsection{Geometric structures} \label{Geometric_structures}
There are several structures that play an important role in KAM theory.
In this Section, we will discuss symplectic and, more briefly,
volume preserving and
reversible systems
(there are other geometric structures that have come to play a role
in KAM theory, but we will not discuss them here).
In this Section, the emphasis will be on the geometric structures and not on
the differentiability properties, so we will assume that vector fields
generate flows, for which variational equations are valid,
etc. (i.e., that they have some mild differentiability properties).
Here we will use Cartan calculus of differential forms rather than the
old-fashioned notation.
Since Cartan calculus uses only geometrically natural operations, it is
conceptually simpler.
This is a great advantage in mechanics, where one frequently uses changes
of variables, restriction to submanifolds given
by regular values of the integrals of motion, etc..
The traditional notation --- in which one writes functions as functions of
the coordinates, e.g., $H(p,q)$ --- is perfectly adequate when the
coordinates are fixed.
On the other hand, when one changes coordinates, one has to decide whether
$H(p',q')$ denotes the same function of new arguments or whether $H(p',q')$
is a different function of $p'$ and $q'$ which produces the same numerical
value as the old function $H$ produced with the old variables $p$ and $q$.
The ambiguity increases enormously when one needs to compute partial
derivatives --- a great deal of the complications in traditional books and
papers on mechanics and thermodynamics arises from this.
For KAM theory these considerations are not so crucial because many of the
operations one has to perform require using Fourier coefficients and the like,
which forces the fixing of a certain system of coordinates.
Nevertheless, we think the conceptual simplification
provided by the geometric notation is worth the effort required in
introducing it.
Now let us start with some important definitions.
\begin{definition}
A {\em symplectic structure\/} in a manifold is given by a 2-form
$\omega_2$ satisfying the conditions
\begin{itemize}
\item[i)] $\omega_2$ is non degenerate
\item[ii)] $\omega_2$ is closed, i.e., $\d\omega_2=0$.
\end{itemize}
A {\em volume form\/} in a manifold of dimension $n$ is an $n$-form
$\omega_n$ that satisfies
\begin{itemize}
\item[i$'$)] $\omega_n$ is non degenerate.
\end{itemize}
\end{definition}
Naturally, an $n$-form $\omega_n$ in an $n$-dimensional manifold
automatically satisfies
\begin{itemize}
\item[ii$'$)] $\d\omega_n=0$.
\end{itemize}
Much of the geometric theory goes through just under the conditions i) and
ii) --- or i$'$) and ii$'$).
When we do not need to distinguish between the symplectic and the volume
preserving cases, we will use $\omega$ to denote either
$\omega_2$ or $\omega_n$.
Properties i) and i$'$) allow us to identify a vector field $v$ with a $1$-
and $(n-1)$-form, respectively, by \mar{identification}
\begin{equation}
\label{identification}
\i_v \omega_2:=\omega_2(v,\cdot)=\gamma_1\ ,\qquad \i_v \omega_n =\gamma_{n-1}\ .
\end{equation}
We will denote the identifications \eqref{identification}
by $\I_{\omega_2}$ and $\I_{\omega_n}$, respectively.
Fundamental examples of a symplectic form $\omega_2$ on $\RR^k\times \RR^k$
and a volume form $\omega_n$ on $\RR^n$ are \mar{examples}
\begin{equation}
\label{examples}
\begin{array}{rcl}
\omega_2 & = &\ds \sum_{i=1}^k \d p_i \wedge \d q_i\ ,\\
\noalign{\vskip6pt}
\omega_n & = &\ds \d x_1 \wedge \ldots \wedge \d x_n \ .
\end{array}
\end{equation}
\begin{remark}
The name symplectic seems to have been originated as a pun
on the name complex. Indeed, there is a sense in which
symplectic geometry is a complexification of
Riemannian geometry. This is actually quite deep
and there is a wonderful new area of research using
methods of complex analysis in symplectic topology.
Since these notes are focused on
KAM theory, it suffices to note that
in mechanics one often finds
the matrix $J \equiv
= \left(
\begin{array}{cc}
0 & \Id_d \\
-\Id_d & 0
\end{array}
\right)
$
which satisfies $J^2 = -1$ and which, therefore,
is quite analogous to multiplication by
$i$ in complex analysis.
\end{remark}
The identification of vector fields with forms plays a very important role
because it allows us to describe the vector fields whose flow preserves the
structure.
Denote by $\Phi_t$ a family of diffeomorphisms of the manifold
generated by the time-dependent vector field $v_t$, i.e.,
$$
\frac{d}{dt} \, \Phi_t = v_t \circ \Phi_t\ ,\qquad \Phi_0 =\Id\ .
$$
In particular, if $v_t$ is independent of $t$, $\Phi_t$ is a flow:
$\Phi_{t+s} = \Phi_t\circ \Phi_s$.
(Again we recall that in this Section we are assuming the objects to be
differentiable enough, in this case $v_t$ to be $C^1$.)
Using the definition of Lie derivative, Cartan's so called ``magic formula''
to express the Lie derivative
\begin{equation}\label{magicformula}
L_X \gamma = d (\i_X \gamma) + \i_X ( d \gamma),
\end{equation} the
closedness of $\omega$ and the definition of $\I_\omega$ we obtain:
$$
\frac{d}{ds}|_{s = 0} \Phi_{t+s}^{*}\omega
= \Phi_t^* L_{v_t} \omega
= \Phi_t^* ( \d \,\i_{v_t}\omega + \i_{v_t}\,\d\omega )
= \Phi_t^* \, \d \, \I_\omega v_t \ .
$$
Thus, if $\omega$ is invariant under the flow $\Phi_t$ (i.e.,
$\Phi_{t}^*\omega = \omega$), we conclude that $\I_\omega v_t$ is closed.
The above result is quite interesting because the $\Phi_t$-invariance
of $\omega$ seems at first sight to be a non-linear and non-local
constraint for the flow $\Phi_t$.
The vector field $v_t$ is perfectly linear and local.
Of particular importance for KAM
theory are the vector fields (called {\em exact
symplectic}, resp.\ {\em exact volume preserving})
for which $\I_\omega v_t$ is exact, i.e.,
$$\I_\omega v_t = \d\gamma_t$$
with $\gamma_t$ a function (symplectic case) or an $(n-2)$-form
(volume preserving case). Sometimes these are called
{\sl globally Hamiltonian} vector fields to indicate
that one can find a Hamiltonian that generates them
globally and not only locally. All
the flows that preserve the symplectic or volume
structure can be expressed locally
as a Hamiltonian flow, but perhaps not globally. We will come back to this in more
detail when we consider some extra structure of the space.
Of course, when we are considering local problems, by Poincar\'e's lemma,
we do not need to distinguish between symplectic and exact symplectic
vector fields.
In the symplectic case, for \eqref{examples},
we have that $\I_{\omega_2}v \equiv \i_v\omega_2 = - \d H$
reduces to the standard Hamilton's equations
$$
v_{p_i} = - \frac{\partial H}{\partial q_i}\ ,\qquad
v_{q_i} = \frac{\partial H}{\partial p_i}\ .
$$
The function $H$ is called the {\em Hamiltonian}
of the vector field $v$. Vector fields satisfying locally
$\i_v\omega_2 = - \d H$ for some function $H$
are called {\em locally Hamiltonian vector fields}.
If the function $H$ can be defined globally,
the vector field $v$ is called {\em globally Hamiltonian}.
An important consequence
of the preservation of symplectic or volume form
is that if a diffeomorphism
$f$ preserves the form $\omega$ and $\i_X\omega = - \d H$,
we have
\begin{equation}
\label{canonicaltransformation}
\i_{f_*X}\omega = \i_{f_*X}f_*\omega = f_* (\i_X \omega) =
- f_* \d H= - \d f_* H = - \d(H\circ f^{-1}) \ ,
\end{equation}
so that $f_*X$ is also a Hamiltonian flow for $H\circ f^{-1}$.
In old fashioned language, this was described as saying that ``canonical
transformations preserve the form of Hamilton's equations'' or some similar
sentence. (In old fashioned books the name canonical
transformations referred to diffeomorphisms preserving
the symplectic form, or sometimes to what we have
referred to as exact symplectic.)
The importance of the
formula \eqref{canonicaltransformation}
is that to make canonical changes of variables
to a Hamiltonian vector field, it suffices to make changes of variables in
the Hamiltonian functions.
This is conceptually much simpler and computationally more efficient.
As we will see, canonical perturbation theory owes its success to this remark.
Note that this calculation goes through both for symplectic and volume forms.
Using Cartan calculus, it is possible to develop perturbation theories for
symplectic and volume preserving flows which are completely analogous.
Notice that in 2 dimensions the volume form and the symplectic structure
are the same and that, when $n=2k$,
$$\omega_n^{\wedge k} :=
\omega_2\wedge \omega_2\wedge\cdots\wedge \omega_2\qquad
\mbox{($k$ times)}$$
is a volume form.
Clearly, a flow that preserves $\omega_2$, also
preserves $\omega_n^{\wedge k}$.
This fact is usually referred to in mechanics
as {\em Liouville's theorem} and is of
fundamental importance since it makes a connection of
mechanics with ergodic theory. Indeed, ergodic theory
was introduced in the study of the relations of this observation
with statistical mechanics.
In the study of Hamiltonian flows,
it is also of interest to study the form
$\mu_E$ defined in the regular
energy surfaces $\Sigma_E = \{ H = E\} $ --
assumed that $d H $ is not degenerate so that it is
an smooth manifold --
by $\omega^{\wedge k} = \mu_E \wedge d H $.
Since $H$ is invariant under the flow, so is
$d H$ and $\mu_E$ is invariant.
The intermediate forms, $\omega_2\wedge\cdots\wedge\omega_2$ ($\ell$ times,
$\ell0$,
so that \eqref{asymptotic} can be used quite quantitatively.
In connection with \eqref{BCH} it is interesting to note that the commutator
of two locally Hamiltonian vector fields is globally Hamiltonian
(see Proposition~\ref{commutator}).
Hence, even if $\L$, $\L^1$ are only locally Hamiltonian, all the $\Tau_n$'s are
globally Hamiltonian and can, therefore, be described by the Hamiltonian
function.
There are several variants of the method of Lie transforms that have been
considered in the literature depending on how we write our candidate map
in terms of exponentials (time-one maps) of Hamiltonian vector fields.
In order of historical appearance some of the methods proposed in the
literature are:
\begin{align}
g_\ep & = \exp (\ep \L_1 + \ep^2 \L_2 + \cdots + \ep^n\L_n+\cdots) \ ,
\label{Deprit}\\[3mm]
g_\ep & = \cdots \exp (\ep^n\L_n) \cdots \exp (\ep^2\L_2)\exp (\ep\L_1) \ ,
\label{Dragt}\\[3mm]
g_\ep & = \cdots \exp\biggl( \sum_{i=2^n}^{2^{n+1}-1} \ep^i \L_i\biggr)
\cdots \exp (\ep^3 \L_3 + \ep^2 \L_2) \exp (\L_1) \ .
\label{LMM}
\end{align}
(See \cite{Deprit69},\cite{DragtFinn},\cite{LlaveMM86} respectively.)
The recursive equation for the perturbation expansions can be computed rather
straightforwardly if we use with abandon --- we can if we interpret the
formulas in the
asymptotic sense --
the formulas
$\exp \L = \sum_{i=0}^\infty \frac1{n!} \L^n$,
think of the $\L^n$ as differential operators and
rearrange the expressions according to the rules of
of non-commutative algebra.
For example, in \eqref{Deprit} we obtain:
\begin{gather*}
\exp (\ep \L_1 \cdots + \ep^n\L_n) H_\ep = H_0\\[2mm]
\L_1H_0 + H_1 = 0\\[1mm]
\left(\tfrac12 \L_1^2 + \L_2 \right) H_0 + \L_1 H_1 + H_2 = 0\\[2mm]
\left[ \tfrac16 \L_1^3 + \tfrac12 (\L_1\L_2+\L_2\L_1)
+ \L_3 \right] H_0 \\[1mm]
\hspace{20mm}+ \left(\tfrac12\L_1^2+\L_2\right) H_1
+ \L_1 H_2 + H_3 = 0 \ .
\end{gather*}
A point that we would like to emphasize is that the equation that we obtain
in the three schemes \eqref{Deprit},
\eqref{Dragt}, \eqref{LMM} for Lie series
is always
\begin{equation}\label{cohomology}
\L_n H_0 + H_n = R_n \ ,
\end{equation}
where $R_n$ is an expression that depends only on previously computed terms.
Using \eqref{Poisson},
we can transform \eqref{cohomology} into
\begin{equation}
\label{cohomology2}
-\H_0 L_n + H_n = R_n \ .
\end{equation}
Note that, if we have a theory for the solutions of equations of the form
\eqref{cohomology2}, we can proceed along the perturbation schemes above.
Note that if we take
$$H_0(p,q) = \omega\cdot p\ ,$$
then \eqref{cohomology2} reduces to the equation \eqref{diff}
that we have studied (under Diophantine assumptions on $\omega$) in
Section~\ref{linear_estimates}. Both the data and the unknown
in \eqref{cohomology2} have an extra variable, but since it enters
as a parameter, we can discuss the regularity of the equation
in terms of the theory that we have developed.
Perhaps more importantly, we note that if we have a good theory of
approximate solutions of \eqref{cohomology2} we can solve the hierarchies of
equations approximately.
This is important in practice as well as in some proofs on KAM theorem.
We also note that an integrable system $H_0(p)$ can be written using
the Taylor expansion
$$H_0(p) = H_0(0) + \omega\cdot p +O(p^2)$$
Hence, we can solve very approximately \eqref{cohomology2} in a sufficiently
small neighborhood of $\{p=0\}$.
This is what is actually used in KAM theory.
These algorithms are also practical tools
that can and have been implemented numerically.
The next two remarks are concerned with some issues about
numerical implementations.
In \cite{LlaveMM86} one can find an appendix
where it is shown that the theories based in
the three schemes above (and in others)
are equivalent in the sense that they give
results which are equivalent in the sense of
asymptotic series.
\begin{remark}
We emphasize that although all the schemes
\eqref{Deprit}, \eqref{Dragt}, \eqref{LMM}
are formally equivalent in the sense that
they require solving the same equations, they are not at all equivalent from
the point of view of
efficiency and
stability of the numerical implementation
or from the point of view of
detailed estimates or even convergence.
As we pointed out, the exponential
of vector fields does not cover any neighborhood of the
origin in the group of
diffeomorphisms so that \eqref{Deprit}
does not provide with a good parameterization
of a neighborhood of the identity and,
perhaps relatedly, it is known to be outperformed in stability etc.\
by (\ref{Dragt})~\cite{DragtFinn}.
The method (\ref{LMM}) \cite{LlaveMM86} is actually convergent in many cases.
Indeed, the KAM theorem asserts it does converge in certain cases
as we will see. For example, it is convergent for the
perturbation series that are based in Kolmogorov's method'
that will be discussed in Section \ref{Kolmogorovmethod}.
The only numerical implementations of \eqref{LMM} that I know of are some
tentative ones carried out by A.~Delshams and the author, but it seems
that the scheme \eqref{LMM}
has a very good chance to be very efficient and stable.
Indeed, it seems to be the only method for which it is possible to
establish convergence.
\end{remark}
\begin{remark}
Sometimes in the numerical solution of the equations
\eqref{Deprit}, \eqref{Dragt}, \eqref{LMM}
it is sometimes advantageous -- both from the point of
view of speed and of reliability --
not to proceed order by order but rather to
take groups of
orders $[2^n, 2^ {n+1} -1]$.
This is tantamount to solving the equations by a Newton method in
the space of families. It has the disadvantage over the
order by order algorithm that at every stage one has to
solve a different equation. This inconvenience is sometimes offset
by the advantage that one linear equation allows one to study many orders
and because the equations that need
to be solved may be more stable than those of other methods.
These quadratic algorithms can be used for all the three methods
described above. Nevertheless, they are somewhat easier to implement
in \eqref{LMM} which has some quadratic convergence already in place.
We emphasize that all the methods can be studied either order
by order or quadratically.
I think that it would be quite important to have a better theory of
these algorithms.
\end{remark}
One lemma that we will be using later is that it is
possible to approximate the action of
the Lie transform on functions by just the first term
in the series of the exponential.
\begin{lemma}\label{leadingorder}
Let $H,G$ be functions on $\torus^n\times \real^n$
endowed with the canonical symplectic structure.
We use the notation of \eqref{analyticnorm} for the
analytic norms of functions.
Assume that:
\begin{itemize}
\item[ i)]
$\|H\|_\sigma $ is finite.
\item[ii)] For a constant $C$ which depends only on the dimension,
we have for $\delta > 0$
\begin{equation} \label{doesnotflow}
\delta^2 > C \|G\|_\sigma \ .
\end{equation}
\end{itemize}
Then, for another constant $\tilde C$ depending only on the dimension,
we have:
\begin{equation}\label{eq:leadingorder}
\| H \circ \exp{ \L_G} - H - \{ H, G\} \|_{\sigma -\delta} \le
\tilde{C} \delta^{-4} \|G\|_\sigma^2 \|H\|_\sigma
\end{equation}
\end{lemma}
\begin{proof}
By Cauchy estimates, \eqref{Cauchyn}, we have:
\begin{equation}\label{doesnotflow2}
\|\nabla G\|_{\sigma -\delta/2} \le \hat C \delta^{-1} \|G\|_\sigma.
\end{equation}
with $\hat C$ a constant that depends only on the dimension.
The constant in \eqref{doesnotflow} is chosen so
that the \RHS of \eqref{doesnotflow2} is smaller than $\delta/2$.
Therefore, all the trajectories of the Hamiltonian flow
generated by $G$ which start in the
region
$$
\D_{\sigma - \delta} \equiv \{ |I| \le e^{2\pi (\sigma - \delta)},
|\Im(\phi)| \le \sigma - \delta \}
$$
do not leave the region $\D_{\sigma -\delta/2}$ for a time
smaller than one (note that they are moving at
an speed that does not allow them to transverse the region
separating the domains in a unit of time).
Hence,
$$
\exp(\L_G) (\D_{\sigma - \delta}) \subset \D_{\sigma - \delta/2}\ .
$$
In particular, we can define the composition $H \circ \exp(\L_G)$
in $\D_{\sigma -\delta}$.
For any point $(I,\phi)$, we can estimate the difference
along a trajectory by using the Taylor theorem with remainder
along a trajectory.
It suffices to estimate the second derivative of
$H$ and the square of the displacement.
The second derivative of $H$ can be estimated
by Cauchy estimates \eqref{Cauchyn}
$ \|\nabla^2 H \|_{\sigma - \delta/2} \le
\tilde{C} \delta^{-2} \|H \|_\sigma$.
The displacement
can be estimated by $\|\nabla G\|_{\sigma - \delta/2}$,
which by Cauchy estimates \eqref{Cauchyn}
can be estimated by $ \tilde{C} \delta^{-1} \|G\|_\sigma$.
Putting these two estimates together, obtains the desired
result.
\end{proof}
\begin{remark}
Analogues of
Lemma~\ref{leadingorder} are true in any analytic
symplectic manifold.
One just needs to define appropriately norms
of analytic functions, Cauchy inequalities, etc.
In the versions of KAM theory
that we will cover in this tutorial, the version we have stated is
enough, but the reader is encouraged
to formulate and prove the more general versions.
\end{remark}
\bigskip
It is also possible to develop a canonical perturbation theory for maps.
Again, the main idea is to change variables so that the system becomes
close to the system which is ``well understood''.
The perturbative equation in this case becomes
\begin{equation}\label{perturbationmap}
g_\ep^{-1} \circ f_\ep \circ g_\ep = f_0 \ .
\end{equation}
We should think of those equations as equations for $g_\ep$ given $f_\ep$.
These equations have been dealt with traditionally by parameterizing
$f_\ep$ using the generating functions method, and similarly for the $g_\ep$.
A more geometric method to use in perturbation theory
is the method of deformations which was introduced in singularity theory.
(In the book \cite{MeyerH92}, one can also find this method
introduced in the Lie transform method.)
It seems particularly well suited to discuss conjugacy equations of a
geometric nature. (See \cite{LlaveMM86, BanyagaLW96} for
some global geometric applications.)
We write
$$
\frac{d}{d\ep} f_\ep = \F_\ep \circ f_\ep \ , \qquad
\F_\ep = \I_\omega (\d F_\ep) \ .
$$
We refer to $f_\ep$ as a family, $\F_\ep$ as the generator and to $F_\ep$
as the Hamiltonian and adopt the typographical convention of
using the same letter to denote the objects associated
with the same family but
using lowercase to denote the family,
calligraphic font to denote the
generator and capital to denote the Hamiltonian.
We note that, under the assumption that $\F_\ep$ is $C^1$, given the generator
and the initial point $f_0$ of the family, we can reconstruct $f_\ep$
in a unique way.
Hence, given $F_\ep \subset C^2$, and $f_0$ we can reconstruct $f_\ep$.
If we express equation \eqref{perturbationmap} in terms of the
generators, it becomes
\begin{equation}\label{perturbationmap2}
-\G_\ep + \F_\ep + f_{\ep *} \G_\ep = 0 \ .
\end{equation}
Expressed in terms of Hamiltonians, it reads
\begin{equation}\label{perturbationmap3}
-G_\ep + F_\ep + f_{\ep *} G_\ep = 0 \ .
\end{equation}
(In the Hamiltonian case, we recall $f_{\ep*} G_\ep = G_\ep \circ f_\ep$.)
There are several advantages in expressing equation
\eqref{perturbationmap} in terms of the generators and the Hamiltonians:
\begin{itemize}
\item
The equations in terms of the generators are linear.
This is natural if we think that the vector fields are infinitesimal
quantities which can, therefore, enter only linearly.
\item
The geometric structure --- not only symplectic, but also volume preserving
and contact (which we have not and will not discuss in these lectures)
are taken care without any extra constraint.
\item
These equations are geometrically natural and can be formulated globally.
\end{itemize}
The proof that \eqref{perturbationmap2} and \eqref{perturbationmap3} are
equivalent to \eqref{perturbationmap} follows easily from the
observation that
\begin{equation}\label{deformationcalculus}
\begin{array}{rcl}
k_\ep & = & f_\ep \circ g_\ep \\[3mm]
\Longleftrightarrow \K_\ep & = & \F_\ep + f_{\ep*} \G_\ep\quad ;
\quad k_0 = f_0\circ g_0\\[3mm]
\Longleftrightarrow K_\ep & = & F_\ep + f_{\ep*} G_\ep \quad ;
\quad k_0 = f_0\circ g_0 \ .
\end{array}
\end{equation}
Even if the equations \eqref{perturbationmap3} is linear in the
Hamiltonian $F_\ep$, we should keep in mind that $f_\ep$ depends on
$F_\ep$ through the very non-linear process of solving the corresponding ODE.
Nevertheless, one can approximate \eqref{perturbationmap3} by
\begin{equation}\label{perturbationapproximate}
F_\ep - G_\ep + f_{0*} G_\ep = 0 \ .
\end{equation}
When
$f_0 (I,\phi) = (I,\phi + \omega)$,
this equation -- for a fixed $I$ --
has the form of \eqref{diff}
the difference equations which were studied in
Section~\ref{linear_estimates}. Since $I$ can be considered
as just a parameter in the data for the equation, we can
use the regularity theory derived for
\eqref{diff}.
If $G_\ep$ is a solution of \eqref{perturbationapproximate}, we note that
\begin{equation}\label{remainder}
F_\ep - G_\ep + f_{\ep*} G_\ep = (f_{\ep*} - f_{0*}) G_\ep \ .
\end{equation}
The intuition is that if $F_\ep$ is small, we can think that $G_\ep$
(obtained by solving a linear equation with $F_\ep$ as \RHS) is small
and that $f_{\ep*} -f_0$
(obtained by solving a differential equation which involves
derivatives of $F_\ep$) is also small.
Hence, the term in the \RHS of
\eqref{remainder} is ``quadratically'' small.
Using the estimates in Lemma~\ref{linearestimates} and mean value theorem etc., we
can prove the estimate in the analytic spaces
$$
\|(f_{\ep*} -f_{0*})G_\ep\|_{\sigma -\delta}
\le C \delta^{-2 \nu -4} \|F_\ep\|_\sigma \ .
$$
Similarly, for the finitely differentiable case,
$$
\|(f_{\ep*} -f_{0*})G_\ep\|_{\Lambda^r}
\le C \|F_\ep\|_{\Lambda^{r+\nu+4}}^2 \ .
$$
Note also that if we write
$$
F_\ep = \ep F_1 + \ep^2 F_2 + \cdots
$$
and try to find
$$
G_\ep = \ep G_1 + \ep^2 G_2 + \cdots \ ,
$$
then \eqref{perturbationmap3}
can be turned into a hierarchy of equations for the $G_n$'s.
All the equations are of the form
$$G_n - f_{0*} G_n + F_n = R_n \ , $$
where $R_n$ is an expression involving previously computed terms.
\begin{remark}
For later developments, it is important to note that
both \eqref{perturbationmap} and
\eqref{perturbationmap3} (and \eqref{perturbationmap2}, \eqref{perturbationmap3})
have a ``group structure''.
This means that if we can find an approximate solution $g_\ep$ (e.g., by
solving the first order equations), we can perform the
\eqref{perturbationapproximate},
\eqref{cohomology2}
change of variables and set
\begin{equation}\label{firststep}
\begin{array}{rcl}
\tilde f_\ep & = & g_\ep^{-1}\circ f_\ep \circ g_\ep\\[2mm]
\tilde H_\ep & = & H_\ep \circ g_\ep\ .
\end{array}
\end{equation}
If we solve the problem for $\tilde{f}_\ep$, $\tilde H_\ep$, i.e.,
\begin{equation}\label{exact2}
\begin{array}{rcl}
\tilde g_\ep^{-1} \circ\tilde f_\ep\circ\tilde g_\ep & = & f_0\\[2mm]
\tilde H_\ep \circ \tilde g_\ep & = & H_0 \ ,
\end{array}
\end{equation}
then, we have solved the original problem since joining
\eqref{firststep} and \eqref{exact2}, we obtain
\begin{equation*}
\begin{array}{rcl}
(g_\ep \circ\tilde g_\ep)^{-1}\circ f_\ep \circ g_\ep \circ\tilde g_\ep
& = & f_0\\[2mm]
H_\ep \circ g_\ep \circ \tilde g_\ep & = & H_0 \ .
\end{array}
\end{equation*}
The importance of the above observation, which will be
appreciated later, is that,
by making successive changes of variables,
we can eliminate all the linear
terms of the error by solving
an equation which is just the linearized
equation {\sl at the integrable system}.
This is an important difference with the standard Newton method
since the standard Newton method requires that we solve
the linearized equation in a
neighborhood.
The fact that we can obtain a method that, for all purposes
is like a Newton method but which nevertheless only requires that
we know how to solve one linearized equation depends crucially
on the fact that the equations that we are studying have a
particular structure which is called {\sl group structure}
and that will be discussed much more in
Section \ref{Implicitfunction}, in particular,
Remark \ref{groupstructure} and Exercise\ref{withoutgroup}.
\end{remark}
\subsection{Generating functions} \label{generating}
One of the reasons why Hamiltonian mechanics is so practical is
because of the ease with which one can generate enough canonical
transformations.
In old fashioned books (\cite{Whittaker}, \cite{Goldstein}, \cite{Landau})
one can find that canonical transformations are described
in terms of generating functions. We will describe those briefly
and only for purposes of comparing with older books. It should be
remarked however, that generating functions, even if
not so useful from the point of view of transformation theory
(there are better tools such as Lie transforms)
are still quite useful tools in the variational
formulation of Hamiltonian mechanics, providing thus a valuable link to
Lagrangian mechanics. Moreover, some of the constructions
that appear in generating functions are quite natural in optics.
See \cite{BornWolf65}.
The equation
$$
f^* \theta - \theta = \d S
$$
is written in old fashioned notations as
\mar{pushforward}
\begin{equation}
\label{pushforward}
p' \, \d q' - p \, \d q = \d S \ ,
\end{equation}
where $p \, \d q := \sum_{i=1}^k p_i \, \d q_i$, etc.
This should be interpreted as saying that we consider
the coordinate functions $p_i$, $q_i$
and the transformed functions $p'_i = p_i \circ f$,
$q'_i = q_i \circ f$. Then, $\theta = p \,\d q$
and $f^* \theta = p' \,\d q'$; $S$ is a function on the manifold.
When $q$, $q'$ are a good coordinate system
(i.e.\ $p$ can be expressed as a function of $q$ and $q'$,
$p=p(q,q')$),
we can define a function $\S:\RR^n \times \RR^n \to \RR$
by setting $\S(q, q'):=S(q,p(q,q'))$. Usually, in old fashioned notations,
this is described as ``expressing $S$ in terms of $q$ and $q'$''
or simply by writing ``$S = S(q, q')$'' or something to that effect.
Very often the same letter is used for $S$, $\S$.
\begin{remark}
In old fashioned notation in mechanics, the same letter is
used for the functions that give the same result
irrespective of the arguments. Of course,
even if this is almost manageable and one understand
what is meant by $S(q,p)$, $S(q,q')$,
by paying attention to the arguments
this notation
wrecks havock when one tries to
evaluate at concrete points.
For example, what is meant by $S(2, \pi)$ when one
is considering at the same time $S(q,p)$, $S(q,q')$?
\end{remark}
Note however, that the assumption that $q$, $q'$
is a system of coordinates is far from trivial.
To begin with, it is not obvious that the manifold
on which we are working admits a system of coordinates.
Even if it does, or if we work just
on a neighborhood so that we have local coordinates,
there are other conditions to be imposed.
For example, it is false for the identity and
for transformations close to identity,
it may be a system of coordinates with undesirable properties.
It is, however, true for $(p,q)\mapsto (p, q+p)$
and small perturbations.
In that case, when we compute the differential in \eqref{pushforward},
we have
$$
\d S = \partial_1 \S(q, q') \, \d q + \partial_2 \S(q, q') \, \d q' \ ,
$$
hence
\mar{graph}
\begin{equation}
\label{graph}
p = - \partial_1 \S(q, q') \ , \qquad
p' = \partial_2 \S(q, q') \ .
\end{equation}
We think of \eqref{graph} as of an equation for $p'$, $q'$
in terms of $p$, $q$. If the implicit function theorem applies
(for which it suffices that $q,q'$ provide a good
system of coordinates on the manifold)
and indeed the equations \eqref{graph} can be solved differentiably, $\S$
determines the transformation.
Note that the implicit function theorem will apply
in a $C^2$ open set of functions $\S$, so that we can think
of this procedure as giving a chart of some subset of the
space of symplectic mappings. Also note that we
parameterize the transformation
by one scalar function.
Moreover, the changes of variables
given by \eqref{graph} are automatically symplectic.
Keeping track of transformations -- in an open set --
which satisfy some non-linear and non-local constraints
(preserving the symplectic structure)
by just keeping track of a function is a great simplification.
However, one important shortcoming of
these generating functions is that for the identity transformation,
$q,q'$ is not a good system of coordinates on the manifold and
we cannot use \eqref{graph} to represent the identity or near
identity transformations. As we have seen, near identity
transformations play an important role in canonical
perturbation theory, so, it is necessary to devise
variants of the method to incorporate them.
In the case that the coordinate functions $p$, $q$ are global
(or that we just work on a neighborhood), we can write
$$
p \, \d q = - q \,\d p + \d (pq) \ .
$$
Hence \eqref{pushforward} reads
\mar{pushforward2}
\begin{equation}
\label{pushforward2}
p' \, \d q' + q \, \d p = \d (S + pq) \ .
\end{equation}
In the case that $p$, $q'$ is a good system of coordinates
(as happens in a neighborhood of the identity), we can write
$$
S + p q = \tilde{\S} (p, q')
$$
and from \eqref{pushforward2} we see that
$$
q = \partial_1 \tilde{\S} (p, q') \ , \qquad
p' = \partial_2 \tilde{\S} (p, q') \ .
$$
Again, we can consider this as a system of implicit equations
defining $p'$, $q'$ in terms of $p$, $q$.
Note that if $q$ is an angle, then
$\S(q+k, q'+\ell) = \S(q, q')$
for all $k, \ell \in \zed^n$. On the other hand,
$\tilde{\S} (p, q'+\ell) = \S(p, q') + p\,\ell$.
Even if this generating function works
in neighborhoods of the identity,
it does not work at all for the map $(p,q) \mapsto (-q, p)$.
One can use similar procedures to obtain
many other generating functions.
For example, one can use for a partition of
$\{1,\ldots,d\} $ into two sets $\A$ and $\B$
the formula:
$$
- \sum_{i\in\B} p'_i \, \d q'_i + \sum_{i\in\A} p_i \, \d q_i
= \sum_{i\in\B} q'_i \, \d p'_i
- \sum_{i\in\A} q_i \, \d p_i
+ \d \left(
- \sum_{i\in\B} p'_i q'_i
+ \sum_{i\in\A} p_i q_i
\right)
$$
to change some of the $p_i$'s for $q_i$'s
in the push-forward.
Even if these procedures are quite customary in old fashioned
mechanics treatises, they will not be very useful for us.
Again, we emphasize that even if the $q,q'$ generating function
can be defined in any exact manifold, the others seem to require
some extra structure, which can be arranged in small neighborhoods.
We note however, that the function $S$ has a well defined
intrinsic meaning as evidenced in \cite{BornWolf65} --- this is sometimes
described as Hamilton-Jacobi equation or ``the action as a function
of coordinates'' depending on what interpretation one gives.
We refer to \cite{Haro00} for much more information on this
primitive function.
In Hamiltonian optics \cite{BornWolf65}, $S$ represents the phase of the wave.
Indeed, Hamiltonian mechanics was developed
as a byproduct of Hamiltonian optics.
This explains why so much of Hamiltonian mechanics,
especially in earlier treatises is based on studying
$S$ and its relatives.
More modern treatments (\cite{Arnold-MathMethods}, \cite{AbrahamMarsden})
prefer to start from the symplectic geometry and postulate it
without any other motivation that it eventually works.
This is certainly expeditious.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Two KAM proofs in a model problem}
In this section we will discuss one of
the technically simplest applications of
the KAM methodology:
the Siegel center theorem.
The main goal of this application is to show
in action perhaps the most basic heuristic principle
of the KAM method:
\begin{quote}
{\large \sf Quadratic convergence can overcome small divisors.}
\end{quote}
Roughly speaking this means that if we have a method of
improvement that reduces the error to something that is
quadratic in the original error, even if
the solution requires solving an equation which involves
small denominators, we can still obtain convergence.
The fact that the convergence does indeed take place
is rather subtle.
In our opinion, the only way to appreciate
the subtlety of the convergence achieved
by KAM theory is to give a serious
try to several other
seemingly reasonable schemes and see them fail.
At the end of the proof, we have suggested several of these schemes
as exercises.
Besides those exercises,
we have also included some exercises which admit easy
solutions and provide extensions to the material in the text.
We also emphasize that the fact that one can get a
quadratically convergent method solving only one
small denominator equation is
far from trivial and it requires that the
equations we consider have some special
structure.
This will be elaborated in more
detail in Section \ref{Implicitfunction}
and in particular in Remark~\ref{groupstructure}.
In this section, we will
present two versions of the Siegel theorem
-- one using just Diophantine conditions in one dimension,
and another using approximation functions and decomposition
in scales in higher dimensions.
The second proof will be formulated as a set of
exercises.
The main ideas of this section
follow \cite{Moser66b}, \cite{Zehnder77} and \cite{Arnold88}.
Indeed, we follow these references rather closely.
These two proofs will illustrate the main features of KAM
proofs and contain the essential analytic and
number theoretic difficulties even if they do not
involve any geometry.
We will start with a one dimensional problem.
See \cite{Moser66b} for more details on the proof
we present and \cite{Zehnder77}
for a higher dimensional version.
\begin{theorem}\mar{Siegel1}
\label{Siegel1}
Let $f: U \subset \cee \to \cee$ be and analytic function of the form\mar{fhat}
\begin{equation}
\label{fhat}
f(z) = a z + \fhat (z)
\end{equation}
with $\fhat(z)=O(z^2)$.
Assume that\mar{diophantines}
\begin{equation}
\label{diophantines}
| ( a^n - 1) ^{-1} | \leq n^\nu K
\end{equation}
and that\mar{smallness}
\begin{equation}
\label{smallness}
\| \fhat \|_1 \leq \rho(\nu) K^2 \ ,
\end{equation}
where $\rho(\nu)$ is an explicit function.
Then there exists a unique function
$$
h(z) = z + \hhat(z)
$$
with $\hhat(z)$ analytic in a disc of radius
$$
\sigma=1-2\rho(\nu)
$$
such that\mar{conjugacys}
\begin{equation}
\label{conjugacys}
f \circ h(z) = h ( a z) \ .
\end{equation}
Moreover, we have\mar{solutionestimates}
\begin{equation}
\label{solutionestimates}
\| \hhat \|_\sigma \leq \|\fhat \|_1 C \ .
\end{equation}
\end{theorem}
\begin{remark}
The uniqueness for $h$ claimed in Theorem
\ref{Siegel1} means that if there are two functions
satisfying this they have to agree in an open set of
the origin. As we have seen already, the condition
\eqref{diophantines} and \eqref{conjugacys} determine
the jet of $h$ uniquely.
\end{remark}
\begin{remark}
Condition \eqref{diophantines} is
automatic when $|a| \ne 1$. In that case, we have presented
a simple proof already. So we will restrict ourselves to the
case when $|a| = 1$.
\end{remark}
\begin{remark}
It is a standard observation that,
assuming that $f$ is defined in a ball of radius~1 and small
is the same as considering a small neighborhood.
Heuristically, in a small neighborhood,
the linear part is the dominant term and it is
natural to try to describe the behavior of the whole system
in terms of the behavior of the linear one.
More precisely, given $f$, consider for $\lambda$ small
$$
f_\lambda = \lambda^{-1} f(\lambda z) \ .
$$
Notice that $f_\lambda$ has the same linear part and is defined in
$\lambda^{-1} B_r$ if $f$ is defined on $B_r$.
Since $| \fhat(z) | = O(|z|^2)$,
we have $\| \fhat_\lambda \|_{B_1} \leq C\lambda$.
If we apply Theorem~\ref{Siegel1} to $f_\lambda$,
we obtain a~$h_\lambda$.
Then $h$ will satisfy \eqref{conjugacys}.
\end{remark}
\begin{remark}
Condition \eqref{diophantines} is not optimal.
Later we will discuss how to obtain the same
result when the arithmetic condition
\eqref{diophantines} is replaced by the Brjuno condition,
which is indeed optimal as shown in
\cite{Yoccoz95},\cite{Perezmarco92}.
The fact that if Brjuno condition
fails one can construct counterexamples is
considerably deeper and out of the scope of these notes. See the references
above.
\end{remark}
Before embarking in the proof, we note that all
the methods are based in estimates for
the equation
\begin{equation} \label{siegelcohomology}
\varphi (a z) - a \varphi(z) = \eta ; \quad \varphi(0) = 0
\end{equation}
in which we consider $\eta$ and $a$ as given and we are
to determine $\varphi$.
The analysis of this equation is very similar
to the analysis of \eqref{diff} in
Section \ref{linear_estimates}.
Since it is not completely identical,
we need to start by revising slightly
the definitions of norms and the setup.
We define the norm of an analytic function by
\footnote{ These norms are slightly inconsistent
with those in Section \ref{prelimanalysis}
in which we took $\|f\|_\sigma = \sup_{ |z| \le e^\sigma} |f(z)|$.
The convention of Section \ref{prelimanalysis} is more natural when one is
using at the same time Fourier series and Taylor series.
For the present section, the convention we now take is more
natural.}
$$
\| f\|_r = \sup_{ |z| \le r} |f(z)|.
$$
\begin{lemma} \label{Siegelanalytic}
Assume that $a$ satisfies \eqref{diophantines}.
Then, if $ \eta(0) \equiv \eta_0 = 0$
we can find a solution of
\eqref{siegelcohomology}.
Moreover
\begin{equation}
\| \varphi\|_{r e^{-\delta}} \le C K |\delta|^{-\tau} \| \eta\|_r
\end{equation}
\end{lemma}
\begin{proof}
This follows from the
results in Section \ref{linear_estimates}.
It suffices to write $z = \exp( 2 \pi i \theta)$.
Then, the result stated is a particular case of
Lemma \ref{linearestimates} applied to
a Fourier series which only has positive
terms.
\end{proof}
\begin{exercise}
Give a direct proof of the Lemma~\ref{Siegelanalytic}
One can follow
the sketch in the beginning of
Section \ref{linear_estimates}.
Start by observing that the
solution of
\eqref{siegelcohomology} is $\varphi_k = \eta_k (a^k -a)^{-1} $.
Estimate
$|\varphi_k|$ using the above formula,
\eqref{diophantines}, and the estimates for
$|\eta_k|$ in terms of $\|\eta\|_r$
obtained using Cauchy estimates.
Estimate $\|\varphi\|_{r e^{-\delta}}$
by the sup of the coefficients.
Then, one ends up with the desired
result with $\tau = \nu +1$.
Since we are dealing with analytic estimates,
this is enough to get through the proof. The ambitious reader
is invited to carry out an analysis similar to
that in \cite{Russmann76} and obtain the optimal
exponent.
\end{exercise}
Now, we proceed to the proof of
Theorem \ref{Siegel1}. The proof we present here
follows \cite{Zehnder77} -- it is a particular case of
the results of that paper.
\begin{proof}
Proceeding heuristically for the moment, we can think of \eqref{conjugacys}
as an implicit equation in a space of functions
$$
0 = \Tau(f,h) \equiv f \circ h - h \circ a
$$
(by $a$ we denote either the constant or the function $a(z)=az$).
Note that $\Tau (a,\Id) = 0$.
We consider $f$ fixed (but close to $a$)
and we are given an approximate solution $h$\mar{initialsituation}
\begin{equation}
\label{initialsituation}
\Tau(f,h) \equiv f \circ h - h \circ a = R \ ,
\end{equation}
where $R$ is the remainder which we would like to think of as small
(the precise sense in which it is small will not be made explicit
in this heuristic discussion).
We would like to obtain a $\Delta$ that eliminates
most of $R$ so that $\Tau(f,h+\Delta) \ll R$.
This amounts to a Newton's method.
Since
$$
\Tau(f,h+\Delta) \approx \Tau(f,h) + D_2\Tau(f,h) \Delta \ ,
$$
we are lead to consider the equation for $\Delta$
\mar{newtongeneral}
\begin{equation}\label{newtongeneral}
R + D_2\Tau(f,h) \Delta = 0 \ .
\end{equation}
In our case -- remember that we are, for the moment,
just proceeding heuristically, but this step is
not difficult to justify -- we have that the derivatives
will be:
$$
D_2\Tau(f,h) \Delta = (f' \circ h) \Delta - \Delta \circ a \ .
$$
Hence, in our case \eqref{newtongeneral} becomes:
\begin{equation}
\label{tosolve}
(f'\circ h) \Delta - \Delta \circ a = - R \ .
\end{equation}
If the factor $f'\circ h = a + \fhat'\circ h$ were just $a$,
the equation \eqref{tosolve} would reduce
to those considered in Lemma~\ref{Siegelanalytic}.
One way that succeeds in reducing the annoying
$\fhat'\circ h $ to a constant is the following:
(in the exercises we examine
several seemingly natural methods
which do not work).
Take derivatives with respect to $z$ of \eqref{tosolve}
and obtain the identity\mar{derivativeremainder}
\begin{equation} \label{derivativeremainder}
f'\circ h \, h' - a h' \circ a = R' \ .
\end{equation}
If rather than looking for $\Delta$, we look for $w$ defined by
$\Delta = h'\, w$ (remember $h$ is close to the identity, so
that indeed $1/h'$ is an analytic function so that
looking for $\Delta$ and for $w$ is equivalent), equation
\eqref{tosolve} becomes
\begin{equation}
\label{tosolve2}
f'\circ h \, h' \, w - h'\circ a \, w \circ a = - R \ .
\end{equation}
Substituting \eqref{derivativeremainder} in \eqref{tosolve2},
we are lead to
\begin{equation}
\label{magic}
a \, h'\circ a \, w - h' \circ a \, w \circ a = - R - R' \, w
\end{equation}
or
\begin{equation}
\label{tosolve3}
a \, w - w \circ a = - (h'\circ a)^{-1} R - (h'\circ a)^{-1} R' \, w \ .
\end{equation}
If we ignore the term $(h'\circ a)^{-1} R' \, w$
(the intuition, which we will later turn into
rigorous estimates,
says that $h'\circ a$ is of order one, $R$ and $R'$ are small,
hence $w$ is small and $R'w$ is much smaller),
we simplify the problem
to studying\mar{constantcoefficients}
\begin{equation}
\label{constantcoefficients}
a w - w \circ a = -(h'\circ a)^{-1} R \ ,
\end{equation}
which indeed is an equation of the type we considered
in Lemma~\ref{Siegelanalytic}.
Hence, the prescription
that we have derived heuristically
to obtain a more approximate solution
is:
\begin{enumerate}
\item
Take $w$ solving \eqref{constantcoefficients};
\item
Form $\Delta = h' w$;
\item Then,
$h+\Delta$ should be a better solution to the problem.
\end{enumerate}
Now, we turn to making all the previous ideas
rigorous. We will need to show that the procedure
improves (that is, show estimates for the
remainder after one step given estimates on the
remainder before starting). We will also need to
show that the procedure can be repeated infinitely
often and that it leads to a convergent procedure.
If we are given a system with an remainder and
run the procedure outlined above,
the following lemma will establish bounds for the new remainder
in terms of the original one.
We will follow standard practice in KAM theory and denote by $C$
throughout the proof constants that depend only on the dimension
and other parameter which are fixed in our proof.
In our case, since we are paying special attention
to the dependence of the domain loss parameter on the size
of the Diophantine constants and the smallness assumptions,
$C$ will not depend on them.
Other KAM proofs which emphasize other features may allow $C$
to stand for constants that could also depend on the
Diophantine constants.
\begin{lemma}\mar{iterativeSiegel1}
\label{iterativeSiegel1}
Let $f$ be as in Theorem \ref{Siegel1}, $h(z) = z + \hhat(z)$,
($\hhat(z) = O(|z|^2)$) defined in a ball of radius
$\frac12 < \sigma < 1$ satisfy
\mar{hypothesis2}
\begin{equation}\label{hypothesis2}
\| \hhat' \| _\sigma \leq M \le 1/2 \ .
\end{equation}
with\mar{hypothesis1}
\begin{equation}
\label{hypothesis1}
\sigma + M < 1 \ ,
\end{equation}
\mar{initialremainder}
\begin{equation}\label{initialremainder}
\| f \circ h - h \circ a \|_\sigma \leq \ep \ .
\end{equation}
Assume furthermore that $\delta > 0 $ is such that
\begin{equation}
\label{hypothesis3}
K C \delta^{-\nu-1} \ep + \sigma e^{-\delta} < \sigma \ .
\end{equation}
Then, the prescription
above can be carried out and we have:
\mar{conclusion1}
\begin{equation}
\label{conclusion1}
\| f \circ (h+\Delta) - (h+ \Delta) \circ a \|_{\sigma e^{-\delta}}
\leq K C \delta^{-\nu-1} \ep^2 + 2 \| f \|_1 (KC\delta^{-\nu-1}\ep M)^2 \ .
\end{equation}
\end{lemma}
\begin{remark}
Notice that since for $\delta \ge 0$,
$
\sigma (1-e^{-\delta}) \leq \sigma \delta \ ,
$
condition \eqref{hypothesis3} is implied by
\begin{equation}
\label{hypothesis2new}
\sigma \delta \geq C K \delta^{-\nu-1} \ep \ .
\end{equation}
which, once we have $\sigma$, just tells us that $\delta$
cannot be smaller than a power of $\ep$.
\end{remark}
\begin{remark}
Note that if we assume without loss of generality that
$\|\fhat\|_1\leq 2$, $K\leq K^2$, $\delta<1$,
the \RHS of \eqref{conclusion1} is less or equal to
\begin{equation}
\label{conclusion2}
C K^2 \ep^2 \delta^{-2(\nu+1)} \ .
\end{equation}
\end{remark}
\begin{proof}
To check that the prescription can indeed be carried out,
we just need to check that the function
$f\circ(h + \Delta)$
can be defined.
Hence, our
first goal will be to obtain estimates on $\Delta$
and show that the image of the ball of
radius $r e^{-\delta} $ under $h + \Delta$ is
contained in the domain of $f$. Indeed, the estimates
for the range will allow us also to obtain estimates
for the derivative of $f$ via the Cauchy theorem
which will later prove to be useful.
Then, we will obtain the estimates in
\eqref{conclusion1} and \eqref{conclusion2} provided that
we have suitable estimates on $\|\Delta\|_{\sigma e^{-\delta}}$.
To obtain the estimates on $\|\Delta\|_{\sigma e^{-\delta}}$,
we note that using the Banach algebra property
of the norms and the inductive assumption
\eqref{hypothesis2}, we can bound the \RHS of
\eqref{constantcoefficients} by
$$
\|(h'\circ a) ^{-1} R\|_\sigma \le (1 - 1/2)^{-1} \| R\|_\sigma \ .
$$
By Lemma~\ref{Siegelanalytic} we have that
$$
\| w \|_{\sigma e^{-\delta}} \leq K C \delta^{-\nu} \|R\|_\sigma \ .
$$
By Cauchy estimates, (see Lemma~\ref{Cauchyn}, but take into
account that now we are in an slightly different situation), we have:
\mar{hcauchy}
\begin{eqnarray}
\| h' \circ a \|_{\sigma e^{-\delta}} &\leq& K \delta^{-1} \|h\|_\sigma\ ,
\label{hcauchy}\\
\| R' \|_{\sigma e^{-\delta}} &\leq& K \delta^{-1} \ep \ .
\label{Rcauchy}
\end{eqnarray}
Hence, taking into account
\eqref{banachalgebra}, and that we had called
$\|R\|_\sigma = \ep$, we obtain from the previous results:
\begin{eqnarray}
\| \Delta \|_{\sigma e^{-\delta}} &\leq& K C \delta^{-\nu-1} \ep M \ ,
\label{deltaestimates}\\
\| R' w \|_{\sigma e^{-\delta}} &\leq& K C \delta^{-\nu-1} \ep^2 \ .
\label{firstterm}
\end{eqnarray}
Note that the assumption~\eqref{hypothesis2},
$$
\| h + \Delta \|_{\sigma e^{-\delta}} < 1 \ ,
$$
so that, as claimed, the composition in~\eqref{conclusion1} indeed makes sense.
To obtain the estimates in \eqref{conclusion1},
we consider the
term to be estimated in
\eqref{conclusion1}
and the obvious identity obtained
just by adding and subtracting terms to
it and grouping the result
conveniently.
\mar{newremainder}
\begin{equation}
\label{newremainder}
\begin{split}
f \circ (h+\Delta) &- (h+ \Delta) \circ a \\
& = f \circ h - h \circ a
+ f'\circ h \, \Delta - \Delta\circ a \\
& + [ f \circ (h+\Delta)
- f \circ h - f' \circ h \, \Delta ] \ .
\end{split}
\end{equation}
The first four terms in the \RHS of \eqref{newremainder},
using \eqref{derivativeremainder} and \eqref{tosolve2} amount to:
$$
R + h'\circ a w + R' w - a h'\circ a w =
R' \, w \ .
$$
The term in braces in \eqref{newremainder} can be estimated because,
by a calculus identity (Taylor theorem with the Lagrange form of
the remainder)
\begin{equation}
\begin{split}
\label{taylorformula}
f(h(z)+\Delta(z))& - f(h(z)) - f'(h(z)) \Delta(z) = \\
& \qquad = - \int_0^1 (s-1)
\,f''(h(z)+s\Delta(z))\,\Delta^2(z)\,ds \ .
\end{split}
\end{equation}
Since, again by Cauchy bounds and \eqref{hypothesis2} we have
$$
\| f''(h(z) + s \Delta(z)) \|_{\sigma e^{-\delta}} \leq
C \delta^{-2} \|f\|_1 \ ,
$$
we can bound the $\| \mbox{ } \|_{\sigma e^{-\delta}}$ of
\eqref{taylorformula} by
\begin{equation}
\label{secondterm}
1/2 \| f \|_1 (K C \delta^{-\nu-1} \ep M)^2
\end{equation}
If we estimate \eqref{newremainder}
putting together \eqref{firstterm} and \eqref{secondterm},
and remembering the standing assumptions on $M, \|f\|_1$,
we obtain~\eqref{conclusion1}.
\end{proof}
To finish the proof of Theorem~\ref{Siegel1},
we just need to show that if $\| \fhat \|_1$ is sufficiently small,
we can repeat the iterative procedure arbitrarily often
and that we converge to a limit which satisfies~\eqref{solutionestimates}.
We will denote by subindices $n$ the objects
after $n$ steps of the iterative process
(assuming that it can be carried out this far).
For example, $\sigma_n$ will be the domain of definition of $h_n$
and we have $\sigma_{n+1}=\sigma_n e^{-\delta_n}$.
To simplify the discussion,
we will use the condition~\eqref{hypothesis2new}
which implies \eqref{hypothesis2} and the bounds \eqref{conclusion2}.
The main thing that we have to do is to choose the $\delta_n$'s.
Notice that if we choose $\delta_n$ going to zero slowly,
we lose more domain than needed and end up with a weaker theorem
--of course, if we lose too fast, we end up with an empty domain.
On the other hand, the smaller that we choose $\delta_n$,
the worse \eqref{conclusion2} becomes.
A reasonable compromise that is neither too fast so
that we end up with no domain nor too slow so that we
can still converge is to choose an exponential rate of
decay. In the exercises, we will explore other choices.
We will choose
\begin{equation}
\label{deltachoice}
\delta_n = \delta_0 2^{-n} \ ,
\end{equation}
and then, will show how to choose $\delta_0$.
With this choice of $\delta_n$, \eqref{conclusion2} implies easily
\begin{equation}
\label{recurrence}
\ep_{n+1} \leq C K^2 \ep_n^2 \delta_0^{-2\mu} A^{2n}
\end{equation}
where $\mu = \nu+1$, $A = 2^\mu$.
We assume by induction that the iterative step can be carried out $n$ times
(i.e., that hypothesis \eqref{hypothesis2new} is verified for the first
$n$ steps).
We will show that, under certain assumptions
on the size of $\delta_0$, $\ep_0$, which will be independent of $n$,
hypothesis \eqref{hypothesis2new} will be verified for $n+1$.
Moreover, we will show that $\ep_{n+1}$ decreases very fast.
Then, by repeated application of
\eqref{recurrence} we have:
\begin{equation} \label{recurrencen}
\begin{split}
\ep_{n+1} &\leq C K^2 \delta_0^{-2\mu} \ep_n^2 A^n \\
&\leq (C K^2 \delta_0^{-2\mu})^{1+2}
\, A^{n+2(n-1)} \,\ep_{n-1}^{2\cdot2} \\
&\leq \cdots \,\,\, \\
&\leq (C K^2 \delta_0^{-2\mu})^{1+2+2^2+\cdots+2^{n-1}\cdot1}
\, A^{n+2(n-1)+\cdots+2^{n-1}} \, \ep_0^{2^{n+1}} \ .
\end{split}
\end{equation}
Note that $1+2+2^2+\cdots+2^{n} \leq 2^{n+1}$
and without loss of generality,
we can assume that $C K^2 \delta_0^{-2\mu} > 1$.
Similarly,
\begin{eqnarray*}
&& n + 2(n-1) + \cdots + 2^{n-1}\cdot 1 \\[1mm]
&& \qquad \qquad = 2^n [n2^{-n} + (n-1) 2^{-(n-1)} + \cdots + 2^{-1}\cdot 1]
\\[1mm]
&& \qquad \qquad \leq 2^n \sum_{k=1}^{\infty} k 2^{-k} = 2^n\cdot 2
= 2^{n+1} \ ,
\end{eqnarray*}
hence
\begin{equation}
\label{iteratedbound}
\ep_{n+1} \leq \left( C K^2 \delta_0^{-2\mu} \ep_0 A \right)^{2^{n+1}} \ .
\end{equation}
Notice that if
$\rho \equiv C K^2 \delta_0^{-2\mu} A \ep_0 < 1$, then
\eqref{iteratedbound} converges to zero extremely fast
(faster than any exponential).
The equation that we need to satisfy
to be able to perform the next step is
$$
\delta_{n+1} \equiv \delta_0 2^{-(n+1)} \geq
C K \delta_0^{-\mu} 2^{-n\mu} \ep_{n+1}
= C K \delta_0^{-\mu} 2^{-n\mu} \rho^{2^{n+1}}
$$
or
\begin{equation}
\label{iteratedcondition}
C K \delta_0^{-\mu-1} \leq 2^{n\mu-(n+1)} \rho^{-2^{n+1}} \ .
\end{equation}
By now, it should be clear that if we take $\delta_0=\frac12$
(so that $\sigma_n\geq e^{-1}$), if we assume that
$\ep_0$ is sufficiently small, we can satisfy \eqref{iteratedcondition}.
Moreover, since by \eqref{deltaestimates},
$$
\| \Delta_n \|_{e^{-1}} \leq K C \delta_0 2^{\mu n} \rho^{2^n} \ ,
$$
we see that
$
\sum \Delta_n < \infty
$
Hence
$$
\Delta \equiv \sum \Delta_n
$$
converges uniformly in the space of
functions in the disk of radius $e^{-1}$
and we can easily bound
$\| \Delta \|_{e^{-1}}$.
\end{proof}
At the end of this subsection, we have collected
some exercises that explore alternatives for the present proof
and for another that will be presented.
Let us highlight some of the remarkable points of the proof.
\begin{remark} \label{wardidentities}
We call attention to the remarkable fact that the
derivatives of
\eqref{tosolve}
could be used to transform the
equation \eqref{derivativeremainder}
into a much simpler equation
(with an error which is small if the remainder
is small and of {\sl quadratic order}).
This is what allowed us to solve the
step with quadratic error. In turn, this
quadratic error was crucial in being
able to deal with the small divisors
(see the following remark).
See exercise \ref{withoutgroup} for an
example of a problem with very similar
analytical properties but without group structure
for which the result is false.
The possibility of performing this
remarkable simplification comes
from the group structure of the equations,
as was emphasized in \cite{Zehnder76}.
This remarkable cancellation has other justifications,
for example, in the context of Lagrangian principles.
Indeed, one can see that it is related to the symmetry
that we used in \eqref{cancellation}. With a bit of
hindsight we can see that the factor
$(1 + {\ell^{[0$?
What happens with $\delta_n = \delta_0 n^{-\alpha}$, $\alpha>1$,
or $\delta_n = \delta_0 n^{-1} (\log n)^{-\alpha}$, $\alpha>1$?
\end{exercise}
\begin{exercise}
Fix $a = \exp\left\{2\pi i \frac{\sqrt{5}-1}{2}\right\}$
and consider
$$
f_N(z) = a z + z ^N \ .
$$
What are the asymptotic of the Siegel radius as $N\to \infty$?
\end{exercise}
\begin{exercise}
\label{withoutgroup}
In the classical Newton method, we use the fact that
if the derivative $D_2{\Tau}(a,\Id) $ is invertible,
then $D_2\Tau(f,h)$ is
invertible when
$(f,h)$ are in a neighborhood
of $(a \Id)$
and, moreover, the norm of the inverse is
bounded.
We can try to apply the same ideas involved
in the proof that in the classical
case the invertibility of the derivative is an open condition
$ (A + B )^{-1} = A^{-1}\sum_{i = 0} (- BA^{-1} )^i $.
(sometimes called the Neumann series)
to
solve the equation
$$
f'\circ h \Delta - \Delta \circ a = -R
$$
by iterating the solution
of
$$
a \Delta - \Delta \circ a = -R - {\hat f}'\circ h \Delta
$$
Try to carry out the procedure and decide whether
it can be applied as an ingredient in a KAM proof.
(e.g. one can try to take more stages in the proof
as one progresses etc.
To the best of the knowledge of
the author it cannot be made to work
(unless one uses cancellations similar to those used in the
quadratically convergent methods or those of the direct methods)
but attempting this will give an
appreciation of the cleverness of the use of
rapidly convergence methods.
Of course, if there is a proof that succeeds in accomplishing this,
the result will be quite interesting.
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hard implicit function theorems.} \label{Implicitfunction}
Before proceeding to more geometric considerations,
it will be convenient to
abstract some of the properties that made the
previous argument work and isolate them in
an abstract implicit function theorem.
This will streamline a good deal of the arguments and
illustrate quite strikingly the principle that the
quadratic convergence can dominate the small divisors.
Even if implicit function theorems take care very
nicely of the analysis of the convergence,
they ignore the geometric considerations and
particularities of the problem at hand.
This particularities are crucial to obtain the general
framework of the implicit function theorem.
Nevertheless, it is useful to introduce the
difficulties one at a time.
Later we will have to spend time making sure
that we can fit a problem or an algorithm to solve
a problem into the functional framework of a
theorem.
We emphasize however that the usefulness of these implicit
function theorems is not restricted to
KAM theory and they have been used in a variety
of problems in geometry, PDE, etc.
and that in any case, they are a very useful strategic
guide on how to organize the proofs of the problem at hand.
There are different versions of implicit function theorems
adapted to work in KAM theory. We just mention
\cite{Zehnder75}, \cite{Hamilton82},
\cite{Hormander90}.(See also \cite{Hormander85}). The main variation we have included
is that we have used the approximation functions
(introduced seemingly in \cite{Russmann80} ) in the implicit
function theorem. Some parts of the
exposition are based on \cite{LlaveV00}.
A very good recent exposition -- regretfully, not easy
to obtain -- of Nash-Moser theorems including
detailed comparisons and examples of applications,
specially to PDE's is \cite{HounieM94}.
Also very important for the relation with PDE's are
\cite{AlinhacG91}, \cite{Hormander90}.
(Of course, one should also consider the work of
\cite{CraigW93}, \cite{Bourgain00}, even if it has not
been formulated as an abstract implicit function theorem
and I am not sure it fits easily into the existing ones.)
The theorem that most closely models the problem
we have discussed so far (and those that we will
discuss later) is that of \cite{Zehnder75},
which he calls {\sl analytic smoothing}
which we now, reproduce,
with an small improvement to deal with
the R\"usmann conditions rather than just the Diophantine conditions.
\begin{remark}
In the case of one-dimensional dynamics, one can use the theory
of continued fractions to show that the R\"usmmann conditions
that we use in the implicit function theorem
(expressed in terms of solvability of
equations) are equivalent to the conditions used in
\cite{Brjuno71} (expressed in terms of number
theoretic properties). I do not know if this equivalence is
true in higher dimensions.
\end{remark}
Note that to abstract the spaces of analytic
functions defined on balls of different radius,
we will consider not just a single Banach space,
but rather a family of Banach spaces.
In the following, it will be good to keep in mind
the proof of Theorem \ref{Siegel1} as motivation
for the definitions and the assumptions.
\begin{theorem}\label{Zehnder}
We will consider scales of Banach
spaces $ \{ X_{\sigma}\}_{\sigma \in [0,1]}$
such that for $
0 \le \sigmap \le \sigma \le 1 $
we have:
\begin{eqnarray}\label{spaceinclusion}
&X_0 \supseteq X_{\sigmap} \supseteq X_\sigma \supseteq X_1 \\
&\| x\|_{X_{\sigmap}} \le \| x\|_{X_\sigma} \label{normdominated} \ ,
\end{eqnarray}
and analogously for $ \{ Y_{\sigma}\}_{\sigma \in [0,1]}$,
$ \{ Z_{\sigma}\}_{\sigma \in [0,1]}$.
Assume that we have $F: X_0 \times Y_0 \to Z_0$
\begin{itemize}
\item[1)] $F(f_0, u_0) = 0 $ for some $f_0 \in X_1, u_0 \in Y_1$.
\item[2)] The domain of $F$ contains the sets
$$
\B_\sigma = \{ (f, u) \in X_\sigma \times Y_\sigma
\ \big| \ \|f - f_0\|_{X_\sigma} \le A,
\ \|u - u_0\|_{X_\sigma} \le B \} \ .
$$
\item[3)] $F(\B_\sigma) \subset Z_\sigma$ and it is
continuous when the range and the domain are given
the natural topologies.
\end{itemize}
In what follows,
$M \ge 1, \gamma > 0, \alpha \ge 0$
will denote
fixed constants.
Assume furthermore:
\begin{itemize}
\item[H1)] $F$ satisfies a so called ``{\sl Taylor estimate}''.
More precisely:
\begin{itemize}
\item[H1.1)] The mapping
\[
F(f,\cdot): Y_\sigma \cap B_\sigma \rightarrow Z_\sigmap
\]
is Frechet differentiable for every $\sigmap < \sigma$.
Denote by $D_2F(f,u)$ the Frechet derivative
and
\begin{equation} \label{Qdefined}
Q(f; u, v) \equiv F(f,u) - F(f,v) - D_2F(f,v)(u-v)
\end{equation}
\item[H1.2)] We have the bounds:
\begin{equation}\label{Taylorestimate}
\| Q(f; u,v)\|_\sigmap \le \Upsilon(\sigma -\sigmap) \|u - v\|_\sigma^2
\end{equation}
where $\Upsilon$ is a decreasing
function. (We will assume without loss of generality
and to avoid complications in algebraic
expressions, that $\Upsilon > 1$.)
The function $\Upsilon$ is called an {\sl approximation function}.
It will also enter in subsequent hypothesis and in (H4) it will be required to
satisfy certain conditions.
\end{itemize}
%\item[H2)] $F$ satisfies a uniform {\sl Lipschitz condition} in its
%first argument. That is, for every $0 < \sigma \le 1$,
%$(f,u), (g,u) \in \B_\sigma $
%\begin{equation}\label{Lipschitzcondition}
%\| F(f,u) - F(g,u)\|_\sigma \le M \| f - g \|_\sigma
%\end{equation}
\item[H3)] {\sl Approximate right inverse}
We can find an approximate right inverse
for the derivative.
That is we can find a linear operator
$\eta$ that maps $Z_\sigma $ into
$X_\sigmap$ for all $ \sigmap < \sigma$ and that
satisfies:
\begin{equation} \label{remainderbounds}
\begin{split}
&\| \eta(f,u) z \|_\sigmap \le \Upsilon(\sigma - \sigmap) \| z \|_\sigma \\
&\| D_2 F(f,u) \eta(f,u) z - z \|_\sigmap \le
\Upsilon( \sigma - \sigmap) \|F(f,u)\|_\sigma \| z \|_\sigma
\end{split}
\end{equation}
\item[H4)] The approximation function satisfies the
{\sl Brjuno-R\"ussmann conditions}:
The function $\Upsilon$ in
\eqref{remainderbounds} satisfies
that there is a sequence $\delta_n > 0$
such that $\sum_n \delta_n = 1/2$,
$\sum 2^{-n} |\log( \delta_n/2 )| < \infty$
and such
\begin{equation} \label{BrjunoRussmann}
\sum_n 2^{-n} \log( \Upsilon( \delta_n) ) < \infty
\end{equation}
\end{itemize}
Then, there exists a constant $C$, depending only on
$M, \alpha$ and $\Upsilon$ such that
if $u_0$ is an approximate solution.
That is:
\begin{equation}\label{approximatesolution}
\|F(f,u_0)\|_1 \equiv \ep
\end{equation}
is sufficiently small, then,
we can find $u^* \in X_{1/2}$ solving exactly
the equation
$$ F(f,u^*) = 0$$
Moreover,
\begin{equation} \label{notmoved}
\|u - u^*\|_{1/2} \le C \|F(f,u_0)\|_1
\end{equation}
\end{theorem}
\begin{remark}
The theorem in \cite{Zehnder75} included also
a hypothesis H2 that allowed one to obtain information on the dependence
of the solutions $u$ in terms of $f$.
We have eliminated the dependence of $u$ on $f$
from the conclusions of the main theorem
and relegated it to remarks
(see Remark \ref{dependence}). Hence, we suppressed H2 from the main
theorem, but kept the
numbering to allow easy comparisons. On the other hand, the
hypothesis H4 here is different from that of \cite{Zehnder75},
but it plays the same role.
\end{remark}
\begin{remark}
There are several equivalent formulations of
hypothesis H4). For all practical purposes, it suffices
to take $\delta$ a fixed exponential sequence.
See the exercises.
\end{remark}
The proof of this theorem is very simple
since we have abstracted away many of the complications
of the previous theorem.
We will present it and then, we will highlight some of the
subtle points and indicate some of the applications.
\begin{remark}
One of the important features of
the proof in \cite{Zehnder75}, which we have
eliminated for this pedagogical presentation,
is that the final result is
expressed in a form which is independent of the
space considered. This requires one to
assume that
$\Upsilon(t) = C t^{-\alpha}$ for some positive $C$, $\alpha$.
This case has
very important consequences such as
the finitely differentiable case.
We will develop these improvements in the
exercises.
\end{remark}
\begin{proof}
We use a quasi-Newton method defined by the iterative procedure
\begin{equation}\label{iterativestep}
u_{n+1} = u_n - \eta(f,u_n) F(f,u_n)
\end{equation}
in which $\eta$ takes the place of the
inverse of the derivative in the regular Newton method.
We set $\sigma_{n +1 } = \sigma_n - \delta_n$ and $\sigma_0 = 1$.
We will obtain recursively estimates of $\|F(f,u_n)\|_{\sigma_n}$
$\|u_n\|_{\sigma_{n} } $,
and of $\| u_n - u_{n+1}\|_{\sigma_{n+1}}$.
Since $\sigma_n \ge 1/2$, the later estimates will
imply that $u_n$ converges in $X_{1/2}$.
Adding and subtracting, we have:
\begin{equation}
\label{newtonestimate}
\begin{split}
F( f, u_{n+1} ) &=
F(f, u_{n+1}) - F(f, u_n) - D_2F(f,u_n) \eta(f,u_n) F(f,u_n) \\
& + F(f,u_n) + D_2 F(f, u_n)\eta(f,u_n) F(f,u_n) \ .
\end{split}
\end{equation}
We can estimate the terms in
the second line in \eqref{newtonestimate} using the
second part of \eqref{remainderbounds}
\[
\| F(f,u_n) - D_2 F(f, u_n)\eta (f,u_n) F(f,u_n)\|_{\sigma_{n+1} }
\le
\Upsilon(\delta_n ) \|F(f,u)\|_{\sigma_n}^2 \ .
\]
Using the first part of \eqref{remainderbounds}, we obtain:
for $\tau_n = (\sigma_n + \sigma_{n+1} ) / 2$
\begin{equation} \label{incrementestimate}
\|\eta(f, u_n) F(f,u_n) \|_{\tau_n}
\le \Upsilon(\delta_n/2) \| F(f,u_n)\|_{\sigma_n} \ .
\end{equation}
This estimate allows us to apply \eqref{Taylorestimate}
to the terms in the first line of \eqref{newtonestimate}.
Hence, we obtain
(bounding $\Upsilon( \delta_n/2 ) > \Upsilon(\delta_n) $)
\begin{equation}\label{stepestimate}
\|F( f, u_{n+1} ) \|_{\sigma_{n+1}} \le
2 \Upsilon( \delta_n/2)^2 \|F(f,u_n)\|_{\sigma_n}^2 \ .
\end{equation}
If we iterate \eqref{stepestimate},
we obtain
\begin{equation}
\label{nestimate}
\begin{split}
\|F( f, u_{n+1} ) \|_{\sigma_{n+1}} \le &
2 \Upsilon( \delta_n/2)^2\times (2 \Upsilon(\delta_{n-1}/2)^2 )^2
\times \cdots \times \\
& \times (2 \Upsilon( \delta_0/2 )^2)^{2^{n} }
\|F(f,u_0)\|_{\sigma_0}^{2^{n+1}} \\
= & 2^{1+ 2+ \cdots + 2^n}
\Upsilon( \delta_n/2)^2 \times
\Upsilon((1/2) \delta_{n-1})^{2^2} \times \cdots \times \\
& \times \Upsilon((1/2) \delta_0)^{2^{n+1}}
\|F(f,u_0)\|_{\sigma_0}^{2^{n+1}} \ .
\end{split}
\end{equation}
We can estimate the logarithm of the factor of
$\|F(f,u_0)\|_{\sigma_0}^{2^{n+1}}$
in the \RHS of \eqref{nestimate} by:
\[
\begin{split}
2^{n+1}\big[ \log(2) +
\log &\Upsilon( (1/2) \delta_n ) 2^{-n} +
\cdots + \log \Upsilon( (1/2) \delta_{n-1} ) 2^{-(n -1)} \\
&
+
\cdots
+
\log \Upsilon( (1/2) \delta_{0} ) 2^{0} \big] \ .
\end{split}
\]
We see that under our assumption H4)
(see \eqref{BrjunoRussmann} ),
the term in braces can be bounded by a constant
(the sum of the series). Hence
\eqref{nestimate} yields
\begin{equation} \label{quadraticconvergence}
\|F( f, u_{n+1} ) \|_{\sigma_{n+1}} \le
(A \|F(f,u_0) \|_{\sigma_0} )^{2^{n+1}}
\end{equation}
where $A$ is a constant depending on the
properties of the approximation function and
the other constants involved in the set up
of the problem.
We see that, if we
$\|F(f,u_0)\|_{\sigma_0} $ is
sufficiently small, the right hand
side of \eqref{quadraticconvergence}
converges to zero extremely fast.
Using \eqref{incrementestimate},
we have:
\begin{equation} \label{nincrement}
\|u_n - u_{n+1} \|_{\sigma_{n+1}} \le
\Upsilon( \delta_n/2)
(A \|F(f,u_0) \|_{\sigma_0} )^{2^{n}}
\end{equation}
where $A$ is also a constant depending
only on the properties of the approximation
function and the other constants involved in the set
up of the problem. (It will be different from the
$A$ in \eqref{quadraticconvergence}, but we
follow the standard practice of denoting all such
constants by the same letter.)
The \RHS of \eqref{nincrement}
is a convergent series because
by our assumption \eqref{BrjunoRussmann}
the general term of the series is bounded.
Therefore,
$\log \Upsilon( \delta_n/2) 2^{-n} \le B$
(where, again, $B$ is another constant depending only
on the constants of the problem and the approximation
function).
When $A \|F(f,u_0) \|_{\sigma_0} < 1$,
the second factor converges to zero faster than
any exponential.
Note also that, if $\|F(f,u_0) \|_{\sigma_0} $ small enough,
the series obtained summing \eqref{nincrement} has a sum
as small as desired. In particular, we can verify that the
limit is close to $u_0$ in $X_{1/2}$.
Hence:
\[
\Upsilon( 1/2 \delta_n)
(A \|F(f,u_0) \|_{\sigma_0} )^{2^{n}} \le
(A e^B \|F(f,u_0) \|_{\sigma_0} )^{2^{n}}
\]
This establishes the claim.
\end{proof}
\begin{exercise}
Many classical proofs of the
classical implicit function theorem are based not in the
Newton method, which is quadratically convergent,
but rather in a contraction mapping principle
(which is called linearly convergence since
the remainder after one step is only a
fixed factor smaller than the remainder
before the step.)
Can one base a method that beats small denominators
on a linearly convergent procedure?
Similarly, one can get algorithms whose convergence is
faster than
quadratic. (For example, solving the equation given by the
second order Taylor expansion or interpolating several
of the previous steps of the algorithm.) Can one
base a hard implicit function theorem on these algorithms?
\end{exercise}
It is interesting to check how the
previous result compares with the
proof we have presented of Theorem \ref{Siegel1}.
The scales of spaces are just spaces of
analytic functions on balls of different
radii. The approximate inverse corresponds to
the solving of the linearized equation
by comparing it with the equation obtained
by taking derivatives of the remainder.
Checking that the scales map into each other
is roughly the same as our inductive hypothesis.
In the presentation of Theorem \ref{Siegel1},
we have, of course taken
\begin{equation}\label{powercase}
\Upsilon(\delta) = M \delta^{-\tau}
\end{equation}
This is a very important particular case of the whole theorem
since it not only appears in interesting situations
but also leads to further consequences which we will discuss
in the following remarks. (Of course, the reader
should also consult \cite{Zehnder75} and the other references.)
The choice of a general $\Upsilon$ satisfying
\eqref{BrjunoRussmann} corresponds to the small
divisors satisfying \eqref{Brjunocondition},
whereas \eqref{powercase} corresponds to
Diophantine conditions.
(For more details see \cite{Russmann90},
\cite{DeLatte97}, \cite{LlaveV00}.)
\begin{remark}\label{groupstructure}
The existence of approximate inverses is a
general feature of conjugacy problems
or of problems having a group structure.
As pointed out in \cite{Zehnder75} p. 133 ff.
existence of
approximate inverses
assuming only the existence of an inverse in the
trivial case
is a general feature of conjugacy problems,
at least at the heuristic level.
This indeed gives a guiding principle for the cancellations that
we found e.g. in the proof of
Theorem \ref{Siegel1} in which we used
that
comparing the
prescription suggested by the heuristic
Newton method with the derivative of
the remainder the linearized
equation suggested by the
heuristic Newton method can be reduced to constant coefficients up to
quadratically small errors.
Notice that the functionals we are solving are
conjugacy equations.
Hence they satisfy the identity
\begin{equation}\label{conjugacyformula}
F( f, u\circ v) = F( F(f,u), v )
\end{equation}
If we take $v = \Id + \hat v$ and we think of $\hat v$ as
infinitesimal, we obtain
\begin{equation}\label{infinitesimalgroup}
D_2 F(f, u) u' \hat v = D_2 F( F(f,u), \Id) \hat v
\end{equation}
If we assume that $\eta D_2( f_0, \Id ) = \Id $,
we obtain that:
\begin{equation}\label{infinitesimalgroup2}
\eta D_2 F(f, u) u' \hat v =
\Id + \eta[ D_2 F( F(f,u), \Id) - D_2F ( f_0, \Id)]
\end{equation}
Notice that we can expect that, if $D_2F$
satisfies some Lipschitz conditions on the
first argument, the term in braces in the \RHS
of \eqref{infinitesimalgroup2} satisfies the bounds we
wanted for an approximate inverse provided
that $\eta$ satisfies the desired bounds.
The importance of this remark is
that by knowing the existence of
$\eta$, which is just an inverse
of $D_2F(f_0,\Id) $ we can deduce, for
functionals with a group structure,
the existence of approximate inverse in a
whole neighborhood, which the hypothesis
needed by Theorem \ref{Zehnder}.
Of course, $F$ satisfies assumption \eqref{conjugacyformula}
when $F(f,u) = u^{-1} \circ f \circ u$ but it could
also be the action by $u$ on vector fields or more complicated
objects and indeed it happens quite frequently when one is considering
geometrical problems.
\end{remark}
\begin{remark}
strategy for KAM (Discussed in more
detail in Section \ref{Kolmogorovmethod})
can be formulated as reducing the
Hamiltonian to a Hamiltonian of a
particular kind.
Hence, we are not interested in
just solving the equation $F(f,u) = 0 $
but rather $F(f,u) = N$, where $N$ is
a submanifold of infinite codimension.
Indeed, this is the problem that is considered
in \cite{Zehnder75} and especially in
\cite{Zehnder76}.
\end{remark}
\begin{remark}
Even if most of the classical KAM problems
(certainly all that will be discussed in this notes)
are conjugacy problems and, therefore, have the group
structure, this is not completely
necessary to have a quadratic algorithm not completely
necessary to have a quadratic scheme.
A review of problems in geometry which are
not conjugacy problems can be found in
\cite{Hamilton82}.
A very interesting recent development is
the observation that variational problems with symmetry
also present another general structure that allows
to obtain quadratic convergence.
See, for example \cite{Kozlov83}, \cite{Moser88}
for PDE's
or, in the context of KAM \cite{SalamonZ89}.
(We will present an account of that work in
Section \ref{Lagrangianmethod}.)
Much more interesting is the fact that in
\cite{CraigW93}, \cite{CraigW94}, another
mechanism to obtain quadratic convergence was
introduced. At the moment, I do not know of
a functional analytic framework that encompasses
these remarkable results.
\end{remark}
\begin{remark}\label{normalform}
In the applications of
the implicit function theorems to problems of
persistence of tori -- and to some geometric problems --
we are not interested in the equation
$F(f,u) = 0$
but rather in the equation
$F(f,u) \in N$ where
$N$ is an appropriate submanifold.
See exercises \ref{ex:normalform},
\ref{ex:Moriyon}
\end{remark}
\begin{remark} \label{consequences}
Note that
the structure of Theorem \ref{Zehnder} is that the input is
just an approximate solution (with some
extra mild requirements) and that the output is
an exact solution not too far from the original
approximate solution.
In the most commonly quoted applications,
the input is the exact solution for
an integrable system, which is an approximate
solution for a quasi-integrable system. Nevertheless,
other applications are possible. Among them,
we mention:
1) Numerical algorithms:
If carefully implemented
and successfully, numerical algorithms produce
approximate solutions (i.e.\ ssomething that, when
plugged into the equations satisfies them approximately).
Hence, using a theorem with the structure of
Theorem \ref{Zehnder}, one
can justify that the approximate solutions
produced by a computer algorithm indeed correspond to
a true solution nearby. In numerical analysis, this is
sometimes called a-posteriori bounds.
(See \cite{BraessZ82}, \cite{LlaveR90}, \cite{CellettiC95}.)
We discuss some numerical issues involved in
Section \ref{compassisted}.
2) Justification of asymptotic expansions (e.g.\ LLindstedt series)
These expansions produce objects that satisfy the
equations approximately. Hence, a theorem
similar to Theorem \ref{Zehnder},
can be used to justify asymptotic expansions.
That is, show that one can indeed find tori
which are not far away from the truncations of
the Lindstedt series. For the KAM tori,
one can find this type of arguments in
\cite{Moser67}. In these case, it is also
shown that the Lindstedt series converge
(since the torus should be analytic as a
function of the parameter).
We emphasize that to justify the asymptotic nature of
the series one just needs that the series
produce objects that satisfy the equation with
smaller errors and that are not too complicated.
The Lindstedt series of lower dimensional
tori are studied by this method in
\cite{JorbaLZ00}. In that case, we do
not know whether the series converges or
not, but following the argument sketched here, it is
possible to show that they are asymptotic in
a certain complex domain.
3) Establishing continuity or
Whitney regularity of the solutions with respect to
parameters -- assuming that $F$ is more regular in
both its arguments --.
This application is
worked out explicitly in \cite{LlaveV00}.
The latter arguments require some certain amount of
uniqueness, which is not provided by the theorem
in the way we have stated and proved it, but which we
obtain in Exercise \ref{ex:uniqueness}.
4) Obtaining a result for finitely differentiable
problems out of the analytic ones.
An application that can already be found in
\cite{Moser66a}, \cite{Moser66b} is that,
as we saw in Lemma \ref{characterization},
we can characterize finitely differentiable functions
by their approximation properties by analytic
functions. We just sketch the argument.
Given a smooth $f$, we study the problem
$F(f,u)$ by considering a sequence
of problems $F(f_n, u_n) = 0$ where $f_n$
are constructed approximating the smooth
function $f$ by analytic functions.
Using that $||f_n - f_{n+1} ||_{2^{-(n+1)}} \le C 2^{-(n+1) r}$
it is often possible (using the structure of $F$)
to show that
$$
||F(f_n,u_n) - F(f_{n+1}, u_n) ||_{2^{-(n+1)}} =
||F(f_{n+1}, u_n) ||_{2^{-(n+1)}} \le C 2^{-(n+1) r'}
$$
We consider $u_n$ as an approximate solution for the problem
with $f_{n+1}$.
In the case that $\Upsilon$ is a power, it follows
that
$$
||u_n - u_{n+1} ||_{2^{-(n+2)}} \le C 2^{-(n+1) r''}
$$
from which, appealing again to Lemma \ref{characterization}
we obtain that there $u = \lim u_n$ which solves
the desired equation and which is analytic.
This method has the advantage that one always works
with analytic functions for which estimates are
often easier and, as we have seen sharper
if one needs to use Fourier coefficients.
We refer to \cite{Moser66a}, \cite{Moser66b},
\cite{Zehnder75}, \cite{Zehnder76} for more details
(such as how to get the induction started),
somewhat different versions of the argument,
and applications to concrete problems.
The quantitative estimates needed to carry out this
strategy are explained in Exercise\ref{improvements}.
5) Bootstrapping the regularity.
A solution which is moderately smooth, if approximated
by an analytic one is an analytic approximate solution.
Applying a theorem of this sort, one can conclude that
given an analytic problem,
if there is a sufficiently smooth solution
(so that the smoothings are indeed very good approximations),
then there is an analytic one. Of course, if one does
have uniqueness of the problem, one obtains that any
solution that has a certain regularity, is analytic.
Of course, if we start with a problem that is very
regular, we can also show that given a solution which is
beyond a certain critical regularity, there will be another
one which is as as the problem allows, and if there is
uniqueness, we conclude that all the solutions beyond
a certain regularity are as smooth as the problem allows.
Arguments of this type are worked out explicitly in
\cite{SalamonZ89}.
Again, we refer to Exercise\ref{improvements} for some
of the quantitative estimates
needed.
\end{remark}
\begin{remark}
Notice that the Theorem \ref{Zehnder} only assumes the existence of
an approximate right inverse.
One should not expect that the solution one produces in the theorem
to be unique. Indeed, in some problems
such as the Nash embedding theorem which motivated a
good deal of the original research
one only has an approximate right inverse and, indeed the solution is
not unique. In many geometric problems, the results we seek
are in any case invariant under diffeomorphisms, so that it
is to be expected that the solution is not unique.
Under moderate assumptions -- e.g. under the existence of
an approximate left inverse -- one gets uniqueness.
See the remarks in \cite{Zehnder75} and see
Exercise \ref{ex:uniqueness}.
These assumptions are often satisfied in KAM theory
or uniqueness of the objects we
are interested in can be obtained by other means.
(Often one seeks geometrical objects in coordinate
systems, so that the geometric objects may be unique
even if their coordinate representation is not.)
One situation when these considerations play a
role is the proof of the KAM theorem following
Kolmogorov's strategy. (See Section \ref{Kolmogorovmethod}.)
In this method, we
seek a change of variables in which the resulting
system manifestly has an invariant torus.
That is, we try to reduce the system to the
the Kolmogorov normal form \eqref{easysystem}.
Such change of variables is manifestly not unique since the
normal form does not specify what are the higher order terms and
one can make changes of variables that only depend to
higher order in the actions. Therefore, one cannot expect
uniqueness in the change of variables nor in the term of
the normal form and a formulation of the theorem based
on this formalism cannot aspire to obtain uniqueness.
Nevertheless, it is true that, under moderate non-degeneracy
assumptions, the torus that has a prescribed frequency is unique.
\end{remark}
\begin{remark} \label{dependence}
The theorem of \cite{Zehnder75} has an
extra hypothesis H2 that requires that $F$
is Lipschitz in the first argument
Then, one obtains Lipschitz dependence on the solution
on the function $f$ (in some appropriate spaces).
We note that, in the case that there is no uniqueness,
the only claim made is that the algorithm \eqref{iterativestep}
leads to a solution that depends in a Lipschitz manner on $f$.
Clearly, when there is no uniqueness
one could make different choices of solutions
for different $f$ and end with a $u$ that depends discontinuously on
$f$.
A detailed treatment of these ideas can be found in
\cite{Zehnder76b}.
We point out that there are other methods to obtain
smooth dependence with respect to parameters that do
not involve following the proof of Theorem \ref{Zehnder}
and checking the differentiability with respect to parameters
of all the steps.
1) One can also obtain quickly higher regularity with respect
to parameters by applying Theorem \ref{Zehnder}
in spaces that consists of smooth families of
functions. Of course, one needs that the approximate
inverse also maps smooth families into smooth families.
This is somewhat tricky since approximate inverses are
not uniquely defined, so one could make different
choices for different values of the parameter and
spoil even continuity.
Nevertheless, for problems with group
structure, the prescription given by \eqref{infinitesimalgroup}
gives a way of accomplishing the solution in spaces of
smooth families of functions.
Arguments of this sort are carried out in detail
in \cite{LlaveO00} to solve a problem in differential geometry.
2) When there is uniqueness, one can follow other sort of
arguments such as finding formal derivatives
for the solution and then, showing that these
formal derivatives satisfy the hypothesis of
Whitney theorem \cite{LlaveV00}.
\end{remark}
\begin{remark}
When one has some regularity -- at least Lipschitz --
with respect to the parameters,
one can start discussing
issues -- important in the applications -- such as the measures
in the space of parameter covered.
\end{remark}
\begin{exercise}
Write precisely the reduction of
Theorem
\ref{Siegel1} to Theorem \ref{Zehnder}
by making explicit choices of spaces, etc.
\end{exercise}
\begin{exercise}
A challenging variant of the previous exercise
is to show that, if the number $\omega$ satisfies the
conditions \eqref{Brjunocondition}, the approximate
inverse we constructed in the proof
of Theorem \ref{Siegel1} satisfies \eqref{BrjunoRussmann}.
If independent study fails,
see \cite{Russmann90}, \cite{DeLatte97} for
estimates that go from arithmetic conditions
to approximation functions.
\end{exercise}
\begin{remark}
In practical applications,
e.g.\ when one is computing numerically solutions
to a problem defined implicitly
one of course, does not compute the inverse of
the matrix, but rather solves numerically the system.
In numerical practice, this usually entails
a factorization of the matrix.
Traditionally, one uses the LU factorization
(Gaussian elimination), even if in KAM theorems
that tend to be ill conditioned one should, perhaps,
prefer the SVD decomposition.
In any case, it is convenient not to have to
recompute these factorizations -- which may much more
costly than the application to a function --.
Of course, we would not like to lose the quadratic convergence
which, e.g. in continuation methods that require great precision is
much more practical that a method that converges more slowly.
The following two schemes,
which avoid having to
recompute factorizations
but which get convergence
faster than linear are studied in
\cite{Moser73} p. 151. The second one
comes from \cite{Hald75}. A geometric
interpretation of these methods
as a Newton method in the space of
jets is discussed in \cite{McGehee90}.
\begin{equation}
\label{quasihald}
\begin{split}
u_{n+1} &= u_n - \eta_{n}F(f, u_n) \\
\eta_{n+1} &= \eta_n - \eta_n(\Id - D_2 F(f, u_{n}) )\eta_n \\
\end{split}
\end{equation}
\begin{equation} \label{hald}
\begin{split}
u_{n+1} &= u_n - \eta_{n}F(f, u_n) \\
\eta_{n+1} &= \eta_n - \eta_n(\Id - D_2 F(f, u_{n+1}) )\eta_n \\
\end{split}
\end{equation}
(In numerical applications, one does not compute the product of
matrices in \eqref{quasihald}, \eqref{hald}. Note that it
suffices to apply the matrices to vectors.)
\end{remark}
\begin{exercise}
Show in finite dimensions that, under smoothness assumptions
and smallness assumptions:
\eqref{quasihald}
leads to
$$
\|F(f,u_{n+1} )\| \le C |F(f,u_n)|^{ (\sqrt{5} + 1)/2}
$$
and \eqref{hald} leads to
$$
\|F(f,u_{n+1} )\| \le C |F(f,u_n)|^2.
$$
\end{exercise}
Applications of these schemes to hard implicit function theorems
and other modifications of the basic algorithm will be
developed in the following exercises.
The following exercises are designed to show that the quadratic
convergence is rather forgiving and that there are many variants
that also work. We have also included some variants in which
the results fail so that the reader can start to develop a feeling
for the range of applicability of the techniques.
\begin{exercise} \label{improvements}
Consider the following improvements to
Theorem \ref{Zehnder}
(either separately or several at the same time, for
the most ambitious reader).
As we will note in the exercises some of them have important
consequences, beyond serving as training.
\begin{itemize}
\item
Modify the hypothesis and the conclusions so
that
the approximate solution is assumed to satisfy
$$
\|F(f,u_0)\|_{\delta_0} \equiv \ep
$$
instead of \eqref{approximatesolution}
and
the conclusion about $u^*$
reads
\begin{equation} \label{notmoved2}
\|u - u^*\|_{\delta_0/2} \le C(\delta_0) \|F(f,u_0)\|_{\delta_0}
\end{equation}
instead of \eqref{notmoved}.
Hint: This result can be deduced from the
statement of the theorem just by a relabeling of the spaces.
\item
Show that in case that we take
$\Upsilon(t) = C t^{-\alpha}$
for some $C, \alpha > 0$, we have
$C(\delta_0) = C' t ^{-\alpha'} $
for some $C', \alpha' > 0$.
\begin{remark}
The previous two items are quite important
since they allow to obtain finite differentiability
out of the analytic result. The strategy to
obtain that is explained in Remark\ref{consequences}.
They are worked out in \cite{Zehnder75}. It can also
be worked out from the statement that we have given
by a rescaling argument.
\end{remark}
\item
Show that in case that we take
$\Upsilon(t) = s \Upsilon_0(t)$
where we consider $\Upsilon_0$ as a fixed function and
$s$ as a variable, the
smallness conditions required in
\eqref{approximatesolution}
are
$$
\epsilon s^2 < K
$$
and that the conclusions \eqref{notmoved}
read
$$
\|u - u^*\|_{1/2} \le K s \|F(f,u_0)\|_1
$$
where now $K$ is a constant depending on all the other properties of
the hypothesis, but independent of $s$.
\begin{remark}
In applications to the KAM theorem, the meaning of the
parameter $s$ is teh allowes size of the Diophantine constant.
This improvement is worked in \cite{Zehnder76b}.
It leads rather directly to estimates on the measure of the
set of tori covered by KAM theorem. (See \cite{LlaveV00}.)
\end{remark}
\item
Consider that in \eqref{Taylorestimate},
\eqref{remainderbounds} we have three different
$\Upsilon$ functions. For example, three different
powers.
(This appears in practice. Some of the
powers come from the Diophantine approximations
whereas others come from the differentiation of composition and the like.)
\item Modify the second equation of \eqref{remainderbounds}
to read
\[
\| dF(f,u) \eta(f,u) z - z \|_\sigmap \le
\Upsilon( \sigma - \sigmap) \|F(f,u)\|_\sigma^{\kappa'} \| z \|_\sigma
\]
for some $\kappa' > 0$.
\item
Modify \eqref{Taylorestimate} to read
\begin{equation}\label{weaktaylor}
\| Q(f; u,v)\|_\sigmap \le
\Upsilon(\sigma - \sigmap) \|u - v\|_\sigma^{1 +\kappa}
\end{equation}
for some $\kappa > 0$.
\item One can also have a different
approximate inverse during the iteration.
\[
\begin{split}
\| dF(f,u_n) \eta_n(f,u_n) z - z \|_\sigmap \le&
\Upsilon( \sigma - \sigmap) \|F(f,u_n)\|_\sigma\| z \|_\sigma \\
& + \exp( - a (1 + \kappa'')^n )
\end{split}
\]
for some $\kappa'' > 0$.
A variant is to choose
\begin{equation}\label{extraquadratic}
\begin{split}
\| dF(f,u_n) \eta_n(f,u_n) z - z \|_\sigmap \le&
\Upsilon( \sigma - \sigmap) \|F(f,u_n)\|_\sigma\| z \|_\sigma \\
& + \exp( - 4^n (\sigma - \sigmap))
\end{split}
\end{equation}
This appears in some proofs (e.g. in Arnol'd type proofs )
when one tries to do some truncation of the problem.
This improvement is not too tricky to do by
itself, but it is not so easy to understand
how it does work with the others.
It is quite enlightening to understand how it works
with the method of obtaining finite differentiability.
with some of the others.
\end{itemize}
Under these modifications, one has to modify slightly the
conditions \eqref{BrjunoRussmann}.
\end{exercise}
\begin{exercise}
Formulate precisely the assumptions of domain loss etc.
to obtain a proof of the implicit function theorem using
an iteration as in \eqref{hald}.
\end{exercise}
\begin{exercise}
Taking into account the improvement suggested in
\eqref{weaktaylor}
give a proof of the theorem
using the scheme of \eqref{quasihald}.
\end{exercise}
\begin{exercise}\label{ex:uniqueness}
Show that if one supplements the
assumption H3 of Theorem \ref{Zehnder}
with the existence of a left approximate
inverse satisfying the same
estimates, one obtains that the solution is
unique in an appropriate sense.
Formulate a precise theorem in which the
domains in which uniqueness holds are
explicitly specified.
(Some version of this is done in \cite{Zehnder75}.)
\end{exercise}
\begin{exercise} \label{ex:normalform}
State and prove an implicit function theorem
in which we do not attempt to solve
$F(f,u) = 0 $ but rather
$F(f,u) \in N$ as explained in
Remark \ref{normalform}.
In that generality, one should not expect uniqueness,
hence, continuity and differentiability with respect to
parameters is presumably not very clean.
\end{exercise}
\begin{exercise} \label{ex:Moriyon}
When considering the normal form problem
one should also modify the
assumption H3 of Theorem
to be:
\ref{Zehnder}
$$
\| dF(f,u) \eta(f,u) z - z \|_\sigmap \le
\Upsilon( \sigma - \sigmap) d_\sigma(N, F(f,u)) \| z \|_\sigma
$$
where $d_\sigma$ denotes the distance between sets
measured with the norm $\| \cdot \|_\sigma$.
This observation appears in \cite{Moriyon82}.
\end{exercise}
\begin{exercise}
A classical theorem in KAM theory is
the theorem of \cite{Arnold61}
which states that given a diffeomorphism
of the circle with a rotation number $\rho$,
which is Diophantine and sufficiently close to
the rotation by $\rho$ in an analytic topology,
then, there is an analytic change of
variables that transforms it in the
rotation by $\rho$.
Formulate it in terms of an abstract implicit
function theorem.
The main difficulty is that, when we
start proving this theorem, we do
not know that the set diffeomorphisms
with rotation number $\rho$ is a manifold.
(We know it after we prove the theorem!.)
Note also that the conjugacy is not unique
since all rotations conjugate a rotation to itself.
I know several ways to do it, but all of them
require some dirty tricks. (A
good source for those -- and for almost anything having to
do with circle maps -- is
\cite{Herman79} and \cite{Herman83}).
Note also that for this problem there are proofs that do
not use KAM. Besides the renormalization proofs
and \cite{KatznelsonO89b}, already mentioned
in Section \ref{newdevelopments}, we mention
\cite{Herman85} and, for constant type numbers (those that have
a bounded continued fraction expansion) \cite{Herman83a}.
\end{exercise}
\begin{remark}
Note that, the estimates we have made to prove
Theorem~\ref{Zehnder} do not use that $\| \cdot \|_\sigma$
is a norm. They would have worked just as well if
$\| \cdot \|_\sigma$ had been a semi-norm.
Of course, in order that the result is meaningful,
we would need that the family of seminorms
$\{\| \cdot \|_\sigma\}_{\sigma\in(0,1/2)}$
defines a useful space, i.e., they define a Fr\'{e}chet space.
See \cite{Hamilton82} for more details about such improvement
and also for applications.
\end{remark}
The original proof of KAM theorems for finite differentiability
were based on different schemes than the proof we have presented.
Note that, for example, the proof of Theorem~\ref{Siegel2}
follows a different scheme.
At every step, the linear operator we have to solve
does have an inverse (not just an approximate inverse).
The problem is that the operator is unbounded
and, hence, simple-minded iterations such as those
of the classical Newton method do not work.
This situation happens also in PDE's.
A notable example was the celebrated Nash embedding
theorem \cite{Nash63}.
The method used in \cite{Moser66a} and \cite{Moser66b}
was to combine steps of the linearized operator with smoothings.
The method allows a norm -- in a space of
somewhat smooth functions -- to blow up,
whereas a norm -- in a space of rougher functions -- decreases.
By using interpolation inequalities,
one can recover good behavior
of some intermediate norms.
(The decrease may not be exactly quadratic,
but it is still is faster than exponential.)
This technique has been highly formalized in \cite{Hamilton82},
which also includes a wealth of applications,
mainly to geometry.
See \cite{Hamilton82}, Section 3, for a comparison
with the methods of Zehnder \cite{Zehnder75}.
In the following, we present a proof along these lines,
which follows rather closely \cite{Schwartz69}.
This book also contains a very nice discussion
of the Nash embedding theorem and on other problems
of nonlinear functional analysis.
In the sequel, we shall refer to a certain range
$m-\alpha \le r \le m+10\alpha$
of spaces $\Lambda_r$ (defined in Section~\ref{prelimanalysis}),
and to a certain constant $M>1$.
We suppose that $M$ is sufficiently large so that the smoothing
operators $S_t$ satisfy
\begin{equation}
\label{smoothingproperties1}
\begin{split}
\| S_t u \|_\rho &\le M t^{\rho - r} \| u \|_r \quad u \in \Lambda_r \\
\| (\Id - S_t) u \|_r &\le M t^{r-\rho} \| u \|_\rho \quad u \in \Lambda_\rho
\end{split}
\end{equation}
for $m-\alpha \le r \le \rho \le m+10\alpha$
($\| \cdot \|_r$ stands for the norm in $\Lambda_r$).
\begin{theorem}
Let $B_m$ be the unit ball in $\Lambda_m$
and $f: B_m \to \Lambda_{m-\alpha}$
be a map that satisfies:
\begin{itemize}
\item[(i)] $f(B_m\cap \Lambda_r) \subset \Lambda_{r-\alpha}$,
for $m \le r \le m+10\alpha$;
\item[(ii)] $f_{\mid B_m\cap \Lambda_r}: B_m\cap \Lambda_r \to
\Lambda_{r-\alpha}$ has two continuous Fr\'{e}chet derivatives,
both bounded by $M$, for $m \le r \le m+10\alpha$;
\item[(ii)] There exists a map
$L:B_m \to {\mathcal B}(\Lambda_m,\Lambda_{m-\alpha})$, where
${\mathcal B}(\Lambda_m,\Lambda_{m-\alpha})$ is the space of
bounded linear operators on $\Lambda_m$ to $\Lambda_{m-\alpha}$ such that:
\begin{itemize}
\item[(ii.a)] $\| L(u) h\|_{m-\alpha} \le M \| h\|_m,
\quad u\in B_m, h\in\Lambda_m$;
\item[(ii.b)] $df(u) L(u) h= h, \quad u\in B_m, h\in\Lambda_{m+\alpha}$;
\item[(ii.c)] $\| L(u) f(u) \|_{m+9\alpha} \le M(1+ \| u \|_{m+10\alpha}),
\quad u\in B_m\cap\Lambda_{m+10\alpha}$.
\end{itemize}
\end{itemize}
Then, if $E:= \| f(0) \|_{m+9\alpha}$ is sufficiently small, there exists
$u\in \Lambda_m$ such that $f(u)= 0$.
\end{theorem}
\begin{proof}
Let $\kappa > 1$, $\beta,\mu,\nu >0$ be real numbers to be specified later.
We will need that they satisfy a finite set of inequalities
relating them and the constants appearing in the assumptions of
the problem.
We construct a sequence $\{ u_n \}_{n\ge 1} \subset \Lambda_m$
by taking $u_0= 0$ and
\[
u_{n+1} = u_n - S_n L(u_n) f(u_n) \,
\]
where $S_n= S_{t_n}$ and $t_n= e^{\beta\kappa^n}$. Later on, we will prove
that this sequence satisfies, for $n\ge 1$:
\begin{itemize}
\item[(p1;n)] $\displaystyle u_{n-1} \in B_m$,
\item[(p2;n)] $\displaystyle \| u_n - u_{n-1} \|_m
\le e^{-\mu\alpha\beta \kappa^n}$,
\item[(p3;n)] $\displaystyle u_n \in \Lambda_{m+10\alpha}$ and
$\displaystyle 1 + \| u_n \|_{m+10\alpha} \le e^{\nu\alpha\beta \kappa^n}$.
\end{itemize}
Notice that, then, $\{ u_n \}_{n\ge 1}\subset \Lambda_m$ converges to some
$u\in\Lambda_m$ and, moreover:
\begin{equation}\label{stepestimate2}
\begin{split}
\| f(u_n) \|_{m-\alpha} = &
\| df(u_n) (u_{n+1}-u_n) - df(u_n) (\Id - S_n) L(u_n) f(u_n) \|_{m-\alpha} \\
\le &
M \| (u_{n+1}-u_n) \|_m +
M^2 t_n^{-9\alpha} \| L(u_n) f(u_n) \|_{m+9\alpha} \\
\le &
M e^{-\mu\alpha\beta \kappa^{n+1}} + M^2 e^{(\nu-9) \alpha\beta\kappa^{n}}.
\end{split}
\end{equation}
Hence, the \RHS of the previous inequality \eqref{stepestimate2}
converges to zero when
$n$ goes to infinity, provided that
\begin{equation}\label{condition1}
\nu<9.
\end{equation}
We are going two prove by induction the three properties satisfied by
the sequence
$\{ u_n \}_{n\ge 1}$. For $n= 1$, condition(p1;1) is trivial.
Condition
(p2;1) reads
\begin{equation}
\begin{split}
\| u_1 - u_0 \|_m = & \| S_0 L(0) f(0) \|_m
\le M t_0^\alpha \| L(0) f(0) \|_{m-\alpha} \\
\le & M^2 e^{\alpha\beta} E \\
\le & e^{-\mu\alpha\beta\kappa},
\end{split}
\end{equation}
where the last inequality holds if
\begin{equation} \label{condition2}
E \le M^{-2} e^{-(1+\mu\kappa)\alpha\beta}.
\end{equation}
Condition (p3;1) reads
\[
\begin{split}
1 + \| u_1 \|_{m+10\alpha} = & 1 + \| S_0 L(0) f(0) \|_{m+10\alpha}
\le 1 + M t_0^\alpha \|L(0) f(0) \|_{m+ 9\alpha} \\ \le &
1 + M^2 e^{\alpha\beta} \le 2 M^2 e^{\alpha\beta} \\
\le & e^{\nu\alpha\beta\kappa},
\end{split}
\]
where the last inequality holds if
\begin{equation}\label{condition3}
1 \le \frac 12 e^{\alpha\beta(\nu\kappa-1} M^{-2},
\end{equation}
that is, if
$\nu\kappa > 1 $ and $\beta$ is sufficiently large.
Suppose now that conditions (p1;j),(p2;j) and (p3;j) are true for $j\le n$.
Then,
\begin{equation}\label{intermediatebound}
\displaystyle \|u_n\|_m \le \sum_{j=1}^{\infty} e^{-\mu \alpha\beta \kappa^j }
\le
\sum_{j=1}^{\infty} e^{-\mu \alpha\beta (\kappa-1) j }=
\frac{e^{-\mu\alpha\beta (\kappa-1)}}{1-e^{\mu \alpha\beta (\kappa-1) j }} <1,
\end{equation}
If we require that
\begin{equation} \label{condition4}
\frac{e^{-\mu\alpha\beta (\kappa-1)}}{1-e^{\mu \alpha\beta (\kappa-1) j }} <1
\end{equation}
which holds when
$$
\mu\beta >> 1
$$
we obtain that the \RHS of \eqref{intermediatebound} is
bounded from above by $1$ and, therefore, we recover (p1:n+1).
To prove (p3;n+1) note that
\[
\begin{split}
1 + \| u_{n+1} \|_{m+10\alpha} &\le
1 + \sum_{j=0}^n \| S_j L(u_j) f(u_j) \|_{m+10\alpha} \\
& \le 1 + M^2 \sum_{j=0}^n e^{(1+\nu) \alpha\beta \kappa^j} \ .
\end{split}
\]
Hence,
\[
\begin{split}
(1 + \| u_{n+1} \|_{m+10\alpha}) e^{-nu\alpha\beta\kappa^{n+1}} &\le
e^{-nu\alpha\beta\kappa^{n+1}} +
M^2 \sum_{j=0}^n e^{(1+\nu-\nu\kappa) \alpha\beta \kappa^j} \\
& \leq 1 \ ,
\end{split}
\]
where the last inequality holds (and so (p3;n+1)) if $\nu>\frac{1}{\kappa-1}$
and $\beta$ is sufficiently large.
Finally, we come to the proof of (p2;n+1). We have:
\[
\begin{split}
\| (u_{n+1}-u_n) \|_m = & \| S_n L(u_n) f(u_n) \|_m \le M^2
e^{\alpha\beta\kappa^n} \| f(u_n) \|_m \\
\le & M^2 e^{\alpha\beta\kappa^n}
(\| f(u_{n-1}) - df(u_{n-1}) S_{n-1} L(u_{n-1}) f(u_{n-1}) \|_m \\
& \qquad +
M \| (u_{n}-u_{n-1}) \|_m^2 ) \\
\le & M^5 (e^{(\nu-9+\kappa)\alpha\beta\kappa^{n-1}} +
e^{(1-2\mu)\alpha\beta \kappa^n})
\end{split}
\]
Therefore, if
\begin{equation}\label{condition5}
M^5 (e^{(\nu-9+\kappa)\alpha\beta\kappa^{n-1}} +
e^{(1-2\mu)\alpha\beta \kappa^n})
e^{-\mu\alpha\beta\kappa^{n+1}}
\end{equation}
we recover (p2;n+1)).
The condition \eqref{condition5} is true when
$\kappa<2$, $\mu> \frac 1{2-\kappa}$, $\nu > 9 - \kappa -\mu\kappa^2$
and $\beta$ is sufficiently large.
Therefore, we have established
that, when the parameters $\mu,\nu,\kappa$ satisfy
\eqref{condition1},
\eqref{condition2},
\eqref{condition3},
\eqref{condition4},
\eqref{condition5} then we can carry out the induction
and establish the theorem.
This is satisfied if we take
$1<\kappa< 2$, $\mu>\frac 1{2-\kappa}$ and
$\frac 1\kappa <\frac 1{\kappa-1} < \nu <9-\kappa-\mu\kappa^2 < 9$.
For instance, $\kappa= \frac 32$, $\mu= \frac{20}{9}$ and $\nu= \frac 94$)
then, choose $\beta$ sufficiently large).
\end{proof}
\begin{remark}
The above methods of proof can also produce results for
$C^\infty$ functions. This is significantly more complicated than the
ideas used so far and we will not discuss them.
\end{remark}
\begin{remark}
In many applications the embeddings of scales of
spaces considered are not just continuous but
also compact. This allows one to improve several of
the steps. See \cite{Hormander90} which also includes
very nice ideas on how to use paradifferential calculus and
several interesting new ideas to obtain very sharp results
on the differentiability.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Persistence of invariant tori for quasi-integrable systems.}
In this section, we will present several
proofs of the theorem that made KAM theory famous.
This theorem is very useful in mechanics and in ergodic theory.
Basically the theorem says that an integrable system
which is not degenerate
(See below for a precise definitions) and
sufficiently differentiable has the property that many
of the quasi-periodic orbits persist under small perturbations.
The theorem has versions for Hamiltonian flows and for
exact symplectic maps.
The simple minded versions that we will discuss
can be stated as follows:
\begin{theorem}\label{KAMflows}
Consider the symplectic manifold
$M = \real^n \times \torus^n $
endowed with the canonical
symplectic form.
Let $H: M \rightarrow \real$
be an analytic function such that:
\begin{equation} \label{hamiltonianperturbed}
H(I, \phi) = h(I) + R(I, \phi)
\end{equation}
Let $\omega \in \real^n$ satisfy \eqref{diophantineflow},
and
$\omega = \nabla h(I_0)$ for some $I_0$.
Assume that for $I$
in a neighborhood of $I_0$ we have:
\begin{equation}
\left|\det \frac{ \partial^2 } {\partial I_i \partial I_j } h(I)\right|
\ge \kappa > 0.
\end{equation}
Then, if $\|R\|_\sigma$ is sufficiently small,
the Hamilton equations for
\eqref{hamiltonianperturbed}
admit a quasiperiodic solution of
frequency $\omega$.
This solution lies on an analytic torus $\Tau$,
which it fills densely. Moreover,
if
$\|R\|_\sigma$ is sufficiently small
$\Tau$ is arbitrarily close
to the torus $\{I_0\} \times \torus^n$
\end{theorem}
The version for exact symplectic maps reads as follows:
\begin{theorem}\label{KAMmaps}
Consider the symplectic manifold
$M = \real^n \times \torus^n $
endowed with the canonical
symplectic form.
Consider the map
$F_0: M \rightarrow M $
given by:
\begin{equation}
F_0(I, \phi) = (I , \phi + \Delta(I))
\end{equation}
where $\Delta: \real^n \rightarrow \real^n $
is an analytic function,
\begin{equation}\label{gradient}
\Delta_j(I) = \frac{\partial}{ \partial I_j } \Phi(I)
\end{equation}
Assume that $\omega \in \real^n$
satisfies \eqref{diophantinemap}
and
$$
\omega = \Delta(I_0)
$$ for some $I_0$ and that
\begin{equation}\label{twistcondition}
\left|\det\frac{\partial}{\partial I_j}\Delta_i(I)\right|\geq\kappa>0
\end{equation}
in a neighborhood of $I_0$.
Let $F: M \rightarrow M$ be an analytic, exact symplectic
map.
If $ \| F - F_0\|_\sigma$ is sufficiently small, then
the map $F$ admits a quasiperiodic
orbit of frequency $\omega$. This orbit is dense in an
analytic torus which (if
$ \| F - F_0\|_\sigma$ is sufficiently small)
is arbitrarily close in the analytic topology
to the torus $\{I_0\} \times \torus^n$ which is
filled densely by the orbit of frequency $\omega$
of $F_0$.
\end{theorem}
\begin{remark}
The condition \eqref{gradient} is imposed so that the
unperturbed map is exact symplectic.
An obvious consequence of \eqref{gradient} is that the matrix
in \eqref{twistcondition} is symmetric.
\end{remark}
In what follows we will indicate several proofs of
the above theorems.
The ideas and techniques of the proofs in both cases are roughly
the same. Moreover, one can pass from one to the other
by an ingenious construction \cite{Douady82}, so that they
are indeed equivalent in a precise sense.
Since proofs of these theorems have been in the literature
for several decades,
and many of the estimates have been covered
in the previous sections, we will leave many of the details to the reader,
indicating the most interesting ones as exercises.
Of course, the theorems, as stated above are quite far from the state of
the art, but we hope that they still contain enough difficulty
to illustrate the techniques of the theory and to fulfill the pedagogical
goal of these notes.
We will also present references to places in the
literature where more elaborate arguments, which
we will try to sketch, lead to
sharper results.
\subsection{Kolmogorov's method.} \label{Kolmogorovmethod}
The original paper has been translated in
\cite{Kolmogorov79}.
A translation of a much less detailed account can be found in
an appendix of \cite{AbrahamMarsden}.
Very good modern implementations of the method
can be found in
\cite{BenettinGGS84}, \cite{Barrar70}.
A generalized discussion of the ideas,
putting them in a much wider context
can be found in \cite{Moser67}, \cite{Zehnder76}.
(See Remark \ref{generalities}.)
We observe that a Hamiltonian system of the form
\footnote{
We use the notation $O(I^2)$
to denote functions $A(I,\phi)$ such that
$A(0,\phi) = 0$, $\frac{\partial}{\partial \phi} A(0,\phi) = 0$
and similarly for other orders
}
\begin{equation}\label{easysystem}
H(I,\phi) = \omega I + O(I^2)
\end{equation}
has Hamiltonian equations of motion
\begin{eqnarray*}
\dot{\phi} &=& \omega + O(I) \\
\dot I &=& O(I^2)
\end{eqnarray*}
Hence
$\phi = \phi_0 + \omega t$, $I=0$
is a solution.
A quasi-integrable system has the form
$$
H(I,\phi) = h(I) + R(I,\phi)
$$
with $R(I,\phi)$ ``small'' in some sense that
will be made precise later.
Clearly we can consider Hamiltonians defined up to constants.
%Notice also that the distinction between $h$ and $R$ is not
%uniquely defined:
%$$
%h(I) + R(I,\phi) = h'(I) + \{[h(I) - h'(I)] + R(I,\phi)\} =:
%h'(I) + R'(I,\phi) \ .
%$$
We will write
$$
h(I) = \omega I + h_2 I^2 + \cdots + h_n I^n + h_{[>n]} (I)
$$
where $h_i I^i$ stands for the homogeneous polynomial of
degree $i$ in the Taylor expansion of $h$
(think of $I^i$ as standing for
all the monomials of degree $i$ ),
and $h_{[>n]} (I) = o(I^n)$.
Similarly, we will write, performing a Taylor expansion in
the variable $I$,
$$
R(I,\phi) = R_0 (\phi) + R_1 (\phi) I + \cdots + R_n (\phi) I^n
+ R_{[>n]} (I,\phi) \ .
$$
Then, we can write the quasi-integrable Hamiltonian as
\begin{equation}
\label{breakup}
H(I,\phi) = R_0 (\phi)+ \omega I + R_1 (\phi) I
+ h_2 I^2 + R_2 (\phi) I^2 + H_{[>2]} (I,\phi)
\end{equation}
We observe that, if
$R_0 (\phi)$ and $R_1(\phi)$ were zero,
we would be in the situation described
in \eqref{easysystem}. To add a bit of color to
the description of the proof, we will refer to these terms
as the ``{\sl bad}'' terms since their presence spoils the
easy argument for existence of quasi-periodic orbits.
The idea of the proof of Theorem \ref{KAMflows}
by this method
is to
find a canonical transformation $C$ -- which will be close to the identity
-- in such a way that $H\circ C^{-1}$ will not have the
bad terms.
The canonical transformation will be constructed as
the limit of a sequence of canonical transformations $C^{(n}$
defined recursively by:
$$
C^{(n+1} = \exp\left(-\L_{G^{(n}}\right)\circ (T^{(n})^{-1}\circ C^{(n} \ ,
$$ where $T^{(n}$ is a canonical transformation of the form
\begin{equation}
\label{translation}
T^{(n}(I,\phi) = (I+k_n, \phi) \ , \qquad (k_n \ {\rm constant}) \ .
\end{equation}
and $\exp\left(-\L_{G^{(n}}\right)$ is the
time one map of the Hamiltonian flow corresponding to the Hamiltonian
$-G^{(n}$. The theory of these canonical transformations
was developed in Section \ref{canonicalperturbation}.
\footnote{
Notice that the canonical transformation in \eqref{translation}
cannot be generated by a time one map of
a vector field generated by a
Hamiltonian function since it is
not an exact symplectic transformation.
One can develop the proof considering the
exponential of a locally Hamiltonian vector
field that combines $\exp\left(-\L_{G^{(n}}\right) \circ (T^{(n})^{-1}$.
(See \cite{BenettinGGS84}.) We prefer to keep the
translations separate with a view to proving
translated curve theorems later.}
We will denote by $H^{(n}$ the Hamiltonian expressed in the
coordinates given by $C^{(n}$. That is,
$H^{(n}\circ C^{(n} = H$. Hence,
$H^{(n+1} = H^{(n} \circ T^{(n} \circ \exp( \L_{G^{(n}}) $.
We will choose the $G^{(n}$ and the $T^{(n}$ in such a way that they
reduce as much as possible the bad terms of the
Hamiltonian $H^{(n}$.
We will try to find the Hamiltonians of these
transformations among linear functions in $I$
\begin{equation}\label{Gform}
G^{(n} (I,\phi) = G_0^{(n} (\phi) + G_1^{(n}(\phi) I \ .
\end{equation}
This is a reasonable choice to try first since this
is the form of the terms that we want to eliminate.
As we will see rather quickly, it works.
(If not, we would have gone back and chosen
a more complicated $G^{(n}$.)
Even if for the proof that we have
discussed here it is enough to verify that the
above form works, the reader that plans to study
new problems may be interested in the fact that
there is a
theory to predict what terms will work and
we have sketched it in Remark~\ref{generalities}.
See also \cite{Moser73} p. 138.
We first describe semi-formally the step to construct
the transformation $G^{(n}$. Since
\begin{equation}\label{transformedhamiltonian}
H^{(n} \exp G^{(n} = H^{(n} + \{ H^{(n}, G^{(n} \}
+ \mbox{``second order in $G^{(n}$''} \ ,
\end{equation}
we try to eliminate the bad terms in the main part of
\eqref{transformedhamiltonian}.
Expanding \eqref{transformedhamiltonian} more explicitly,
taking into account \eqref{Gform} and \eqref{breakup},
we have:
\begin{equation}
\begin{split}
\label{toeliminate}
H^{(n} + \{ H^{(n}, G^{(n} \}
=&\,\,\, \omega I \\
&+ \{ \omega I, G_0^{(n} \} + R_0^{(n} (\phi) \\
&+ \{ \omega I, G_1^{(n}(\phi) I \}
+ \{ h_2^{(n} I^2, G_0^{(n} (\phi) \} + R_1^{(n} (\phi) I \\
& + \{H_{[>1]}^{(n} (I,\phi) , G_1^{(n} (\phi) I \} \\
& + \{H_{[>2]}^{(n} (I,\phi) , G_0^{(n} (\phi) \}
+ H_{[>1]}^{(n} (I,\phi) \\
& + \mbox{``second order in $G^{(n}$''}
\ .
\end{split}
\end{equation}
Notice that the ``bad'' terms of
\eqref{toeliminate}
(i.e.\ those that do not include $I$ or include it
only to the first power)
are precisely those on the
second and third lines
(up to ``second order in $G^{(n}$'' terms).
The goal will be to choose $G^{(n}$ in such
a way that the bad terms in the
resulting Hamiltonian
are much smaller than those in the
original system. If
we manage to
eliminate the bad terms in
the main part of
\eqref{toeliminate}, the
Hamiltonian in \eqref{transformedhamiltonian}
will only have bad terms which are ``second order in $G^{(n}$''.
We claim that it is always possible to find a $G_0^{(n}$
in such a way that we eliminate the bad term with no
powers of $I$.
Equating the second line of \eqref{toeliminate} to zero, we
obtain the following equation for $G_0^{(n}$:
\begin{equation}\label{toeliminate0}
\{ \omega I, G_0^{(n} \} + R_0^{(n} (\phi) = 0
\end{equation}
Equation \eqref{toeliminate0}
is, of the form \eqref{der} for which we have developed a theory
in Lemma \ref{linearestimates}.
Note that $\{ \omega I, G_0^{(n} \} = L_\omega G_0^{(n} (\phi)$
where $L_\omega$ is defined in \eqref{der}.)
The main conclusion of the theory of Lemma \ref{linearestimates}
is that the equation \eqref{toeliminate0} can
always be solved (with an slightly regular function), if
the \RHS has average zero.
Notice that, since Hamiltonians are defined only up to the addition
of a constant, we can always ensure that $R_0^{(n}$ has average
zero and, hence, that equation \eqref{toeliminate0} can be
solved for $G_0^{(n}$.
Eliminating the second bad term in \eqref{toeliminate} is more subtle.
The equation to eliminate this term is
\begin{equation}
\label{secondbad}
\{ \omega I, G_1^{(n}(\phi) I \}
+ \{ h_2^{(n} I^2, G_0^{(n} (\phi) \} + R_1^{(n} (\phi) I = 0
\end{equation}
The $G_0^{(n}$ appearing in \eqref{secondbad} is known since we found it
by solving \eqref{toeliminate0} so that the equation
\eqref{secondbad} is only an equation for $G_1^{(n}$
and all the other terms in it are known.
Noting that all the terms have the structure of the dot product
of a vector (depending on $\phi$) with $I$
and eliminating this vector,
we can write the equation \eqref{secondbad} as
\begin{equation}
\label{secondbadvector}
L_\omega G_1^{(n} (\phi) = - 2 h^{(n}_2 \nabla G_0^{(n} (\phi)
- R_1^{(n} (\phi) \ .
\end{equation}
The equation for each of the components of \eqref{secondbadvector}
is just one equation of the form \eqref{der}.
We see that \eqref{secondbadvector} will have
a solution when and only when the average of
the right-hand side is equal to zero.
The average of the term $h^{(n}_2 \nabla G_0^{(n} (\phi)$
is automatically zero.
Hence, we conclude that if
\begin{equation}
\label{inductiveK}
\int_0^1 R_1^{(n} (\phi) \, d\phi = 0 \ ,
\end{equation}
we can indeed solve \eqref{secondbadvector} and, hence,
eliminate the second part of bad terms.
Of course, \eqref{inductiveK} is very restrictive.
It is very easy to construct perturbations that
do not satisfy the condition.
Here is when the translations $T^{(n}$ come into play.
Given any Hamiltonian of the form
\eqref{breakup}, provided that the
$h_2^{(n}$ satisfies the non-degeneracy assumptions,
it is possible to choose a translation
$T^{(n}$ of the form \eqref{translation} in such a way
that the average of $R_1^{(n}$ vanishes.
This is an application of the implicit function theorem
provided that $h_2^{(n}$ is an invertible matrix
and that, of course, all the $R$ terms are small.
Notice that the ``vertical'' translation by $k_n$
is roughly given by
$k_n \approx - \frac 12 \left(h_2^{(n}\right)^{-1} {\bar R}_1^{(n}$.
(We call attention to the fact that the conditions that
need to be adjusted in \eqref{inductiveK} is
exactly the number of parameters at our disposal
when we apply a translation.)
The magnitude of the translation required to adjust
the average of $R_1^{(n}$ can be bounded by a constant times
the size of $R_1^{(n}$ (provided that $h_2^{(n}$ is invertible
and that the other terms are small, so that we
can apply the implicit function theorem).
\medskip
Hence, the algorithm for the iterative proof is:
\begin{enumerate}
\item
To determine the translation so that
$H^{(n} \circ T^{(n}$ satisfies the normalization
$$
\int_0^1 \frac{\partial}{\partial I} H^{(n} \circ T^{(n} \Big|_{I=0}
\, d\phi = \omega \ .
$$
\item
For the ``new'' $H^{(n}$ (i.e.\ for $H^{(n} \circ T^{(n}$)
find $G_0^{(n}$ and $G_1^{(n}$ in such a way that we eliminate
the two ``bad terms'' in \eqref{expansion}
up to quadratic error.
\end{enumerate}
We have already seen that step~2
involves small divisors and unbounded operators.
Nevertheless, we have also seen several times that
the quadratic convergence can overcome the effect
of small denominators (for Diophantine numbers).
Compared with the previous cases we have
dealt with, the only new complication of
the present algorithm is that
we have to deal with the extra complication
of having to adjust the translation so that
\eqref{secondbadvector} becomes solvable.
The main complication of the translation is
that terms that were high order generate lower order terms.
For example, a ``good term'' $H(I,\phi) = f(\phi) I^2$,
with $f(\phi)$ a $\phi$-dependent quadratic form becomes upon
translation
\begin{equation}
\label{translated}
H\circ T = f(\phi) I^2 + 2 f(\phi) I k + f(\phi) k^2 \ .
\end{equation}
The last two terms of \eqref{translated} are ``bad''.
The fact that find a
translation to eliminate the average in
\eqref{secondbadvector}
depends on the fact that the quadratic
term $h_2^{(n}$ is invertible. We need to keep track of
the fact that this remain so under the successive changes of
variables.
This is not so difficult since the condition is
an open condition.
>From the analytic point of view, we note that the
procedure involves solving (twice) equations of
the form \eqref{der} and applying the implicit function theorem.
As we did in Theorem~\ref{Siegel1}, the {\sl second order} terms
can be estimated in analyticity domains using
Cauchy estimates.
In summary, we have sketched a procedure that
given a perturbation
that satisfies certain non-degeneracy conditions,
makes a change of variable that reduces
the bad terms and
whose resulting error is smaller.
More precisely, given estimates of the bad
terms in a domain, we can obtain estimates of the
resulting bad terms in an slightly smaller domain.
The estimates will be of the form
$ || New||_{\sigma e^{-\delta} } \le C \delta^{-\tau} || Original||_{\sigma}$
Note also that, in order to match domains etc. we need that
$\delta$ and the size of the remainder are suitably related.
The proof consists in showing that
if the original error is sufficiently small,
then we can carry out indefinitely
the iterative procedure sketched above
and it converges
in a non-trivial domain.
Here we sketch the main considerations that need
to be taken into account converting the above
remarks into a proof. The reader is urged to
either work them out alone or to use this
as a reading guide for excellent
expositions in the literature (some of them are
discussed below).
\begin{itemize}
\item[A)]
We start by deciding that
we consider domains loses of the
form
$ \delta_0 2^{-n}$,
and that we will do estimates
on domains parametrized by a
$r_n$ defined by
$r_{n+1} = r_n - \delta_0 2^{-n}$.
\item[B)]
We will need to assume inductively that
\begin{itemize}
\item[B.1)] We have bounds:
$$
\| {( h_2^{(n})} ^{-1} \| \leq C_1 \ ,
$$
and that the derivatives of $R$ are sufficiently small
so that they do not affect the application of the implicit
function theorem (to ensure the existence
of the translation $T^{(n}$).
We take $C_1$ to be twice the initial constant:
$C_1 := 2\, \| {(h_2^{(0})} ^{-1} \|$.
We will need to check that, if the initial error
is small enough, the iterative procedure keeps the assumption being
valid.
\item[ B.2)]
Assume inductively that:
$$
\| R^{(n} \|_{r_n} \leq C_2 \ ,
$$
with $C_2$ being twice the initial value:
$C_2 := 2 \, \| R^{(0} \|_{r_0} $.
\item[B.3)]
We will also assume that we have bounds similar to those
in the study of the Siegel problem
\begin{equation}
\label{domainmatch}
\| \nabla G^{(n} \|_{r_n} \leq \delta_0 2^{-n\tau}
\end{equation}
The goal of the latter bounds \eqref{domainmatch}
is to ensure that when we perform the composition of
$H^{(n} \circ T^{(n} \circ \exp (\L_ {G^{(n}}) $,
the composition is still defined in the smaller domain.
\end{itemize}
\item[C)]
Using assumption B.1, we are able to control the size
of the translation by $\| R^{n)}\|_{r_n}$
times an universal constant.
Given B.2, we see that the size of the remainder of
$H^{(n} \circ T^{(n}$ is still of the same order of magnitude
as~$\|R^{(n}\|_{r_n}$.
(The new lower order terms generated are bounded
by the size of the translation.)
\item[D)]
Solving the small divisors equation, we obtain
$G_1^{n)}$, $G_0^{n)}$. We can bound
$$
\| G_1^{(n} \|_{r_{n+1}} + \| G_0^{(n} \|_{r_{n+1}}
\leq C K^2 \, \frac{2^{n\tau'}}{\delta_0^{\tau'}}
\left( \| R_0^{(n} \|_{r_n} + \| R_1^{(n} \|_{r_n}
\right)^2 \ ,
$$
The factor $2^{{n\tau'}}{\delta_0^{\tau'}}$ is
the usual small divisor factor when we take domain losses
as in A).
\item[E)]
The heuristics can be justified by adding and subtracting
and applying the mean value theorem pretty much in the same way
that we did in the proof of Siegel theorem
but using the estimates we developed
in Section~\ref{canonicalperturbation}.
We obtain:
\begin{equation}
\label{quadraticestimates}
\| R_0^{(n+1} \|_{r_{n+1}} + \| R_1^{(n+1} \|_{r_{n+1}}
\leq C K^2 \, \frac{2^{n\tau'}}{\delta_0^{\tau'}}
\left( \| R_0^{(n} \|_{r_n} + \| R_1^{(n} \|_{r_n} \right)^2 \ .
\end{equation}
\item[F)]
The rest is essentially mopping up:
\begin{itemize}
\item[F.1)]
We need to show that the quadratic convergence implied
by \eqref{quadraticestimates} implies that the inductive assumptions
in B) remain valid (if we start with a small enough error).
This is accomplished in a similar manner as that
in the Siegel theorem (the only delicate one is
\eqref{domainmatch} and this is exactly the same
as in the Siegel domain).
\item[F.2)]
We need to show that the accumulated transformation converge.
Again, this is not very delicate since the quadratic convergence
implies that $C^{(n}$ are converging to the identity
extremely rapidly.
\end{itemize}
\end{itemize}
We urge the reader to compare the above
sketch with the papers
\cite{BenettinGGS84} and with \cite{Barrar70}
which contain very readable full proofs.
The main difference in the strategies of those papers
with the presentation here is that
\cite{Barrar70} uses generating functions to deal
with canonical transformations.
Both of \cite{BenettinGGS84} \cite{Barrar70}
do not make a distinction
between the translations and the exact exact transformations
and they use just one locally hamiltonian
transformation that accomplishes the effect of the two steps
that we discussed.
This is, of course, perfectly fine for the problem at hand.
we have, however, preferred to keep the two types of
transformations
separate with a view in translated curve theorems.
A very pedagogical proof of a particular case of the result
(that nevertheless contains the most essential difficulties) is
\cite{Thirring97}. The paper \cite{Zehnder76} contains
a detailed reduction of the proof based in the Kolmogorov
method to an abstract implicit
function theorem very similar to Theorem \ref{Zehnder}.
\begin{remark}
The Kolmogorov method of proof has the advantage
that it is quite direct and very well suited to functional analysis.
We always deal with the same linearized equation
with the same frequency.
In particular, it leads to very good regularity results.
The main disadvantages arise from the fact that every different
frequencies require different transformations.
Moreover, the form \eqref{easysystem} is not unique.
Natural question, which are important for applications,
but that do not follow directly from the results are
what is the measure covered by the tori and determining
whether tori of similar frequencies are close together.
(Indeed, so far, we have not shown that there is only one torus
with the a given frequency. Note that there are
many hamiltonians with the same form \eqref{easysystem}.)
The question about the measure occupied by tori
can be answered by showing that
the mapping which associates to a frequency $\omega$ satisfying
\eqref{diophantineflow}
the torus with frequency $\omega$ produced
in the Theorem \ref{KAMflows} is Lipschitz.
Moreover, the tori can be expressed as the graph
of a function of $\phi$,
\begin{equation}
I=W_\omega(\phi).
\end{equation}
Clearly, given one torus, there is only one
function $W$, whereas, given one torus, there
will be several hamiltonians of the form
\eqref{easysystem} and several transformations reducing
the original flow to them.
It is true that $W_\omega(\phi)$ turns out to be Lipschitz
with Lipschitz constant close to $\|h_2^{-1}\|$.
The proof of these Lipschitz properties can be obtained
rather easily if we note that the system of the form
\eqref{easysystem} is also an approximate solution
to the equation for $\tilde{\omega}$ in the plane of $\omega$.
Hence, if $\omega$ and $\tilde\omega$ are close enough,
we can consider the torus for frequency $\omega$ as
an approximate solution for the equation
that would produce a torus of frequency
$\tilde \omega$. The error of the approximation is
controlled by $\omega - \tilde \omega$.
Hence, applying the procedure, we see that we produce a
solution which differs not more than something that
can be controlled by $\omega - \tilde \omega$.
This type of argument also leads to uniqueness
results of the torus with a given frequency.
More details on this type of argument can be found
in \cite{Douady88}, \cite{Sevryuk95}.
As we will see later, it is true that the map
$(\omega,\phi)\mapsto W_\omega(\phi)$
introduced above is differentiable in the sense of Whitney.
\end{remark}
\begin{remark}
Another aspect in which the method of proof we have discussed
is not optimal is that it requires very strong
non-degeneracy conditions.
Notice that we want to ensure that
the size of the
translation required to
adjust the error to zero average
is commensurate with the error.
In a degenerate situation, the size of the translation
would be a root of the size of the error and then,
the method as we have presented it, would collapse.
As a matter of fact, one can get a better non-degeneracy
condition if one does not fix the frequency,
but fixes it up to a multiple.
Hence, the only thing that we require is that
${\rm Span}(\omega)+{\rm Range}(h_2)=\RR^n$.
One can also use clever tricks to reduce degenerate situations
to non-degenerate ones. For example,
in \cite{BroerH91}.
As we will see later, one can do significantly better than that
by using other methods. For example, \cite{Sevryuk95}.
\end{remark}
\begin{remark} \label{generalities}
There is an interesting interpretation of the
method of proof we have presented above in terms of
geometry in infinite dimensional spaces.
This interpretation can certainly serve as a heuristic
guide and many KAM theorems can be fit
into this form. It was proposed in \cite{Moser67}
and developed quite forcefully in
\cite{Zehnder76}, which developed
in this language the main KAM theorems.
In \cite{Hamilton82}, a similar philosophy is applied
to many geometric problems.
The idea is to think of \eqref{easysystem} as defining
a manifold $\N$ in the space of Hamiltonians $\H$. All the elements of
this manifold have a feature that we are interesting in
studying. In this case, having
an invariant torus of frequency $\omega$.
We also have an action $\Psi$ of a group. In this case, the action by
canonical transformations.
The proof we have sketched shows that given a neighborhood $\U$
of
$\N$ in $\H$ all the elements of $\U$ have an orbit under
$\Psi$ that intersects $\N$.
Even if this is not completely trivial to make precise,
(one has to define the topologies of the spaces of hamiltonians
and mappings, check
that they are manifolds, check the properties of the action of the group
of transformations on it, etc.)
it can serve as a heuristic principle to decide which theorems
are possible. (Note that, if we were considering a
finite dimensional problem, we could just decide what
was true by deciding whether the tangent spaces of $\N$
and of the orbits of the action span the tangent space of
$\H$. )
We note that this line of reasoning and these
heuristic principles apply to other problems outside mechanics.
Indeed, a good part of singularity theory can be formulated in this
way. Similarly,
many problems in geometry and PDE can be reduced
to implicit function theorems by applying this heuristic picture.
(See \cite{Hamilton82}.)
\footnote{
Incidentally, in singularity theory one has a very
powerful implicit function theorem \cite{Mather69}, which allows
to deal in some cases with operators that loose fraction of
the derivatives.
The method of \cite{Mather69} paper has, to my knowledge not been used in
KAM theory, even if \cite{LlaveMM86}, which
considers perturbations theories for Hamiltonian systems that
were, previously done using KAM theory,
was very inspired by it.}
The idea of deciding which theorems in KAM theory could be true
by just looking at when the tangent spaces span leads very
quickly to the problem of counting parameters.
(See the discussion in \cite{Moser67}.)
Roughly one needs that the normal form $\N$ and the
group acting contain enough free parameters to overcome
all the obstructions imposed by the geometry.
One of the important developments of later years is
that in this counting of parameters, one should include
the frequency \cite{Eliasson88a} or
the perturbation parameter \cite{JorbaS92}.
One reason why this is not obvious is
that these extra parameters have a Cantor structure, hence
at first sight, notions based on the geometry of tangent
spaces etc. do not seem workable.
Nevertheless, it turns out to be true that
one can use these Cantor parameters
very much in the same way as continuous families
supplementing the standard geometric arguments based on
implicit function theorems with measure theoretic estimates.
Indeed, the next method of proof which we discuss can be used
to cope with this type of problems.
We refer to \cite{Sevryuk99} for an account of recent
developments in the lack of parameters problem and,
relatedly on the problem of study of degenerate systems.
\end{remark}
\begin{exercise}
Try to carry out the proof choosing
the translation $T^{(n}$ given by
$k_n = - \frac 12 \left(h_2^{(n}\right)^{-1} {\bar R}_1^{(n}$.
Notice that in such a choice we kill the average of $R_1^{(n}$
up to second order terms in~$R^{(n}$.
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Arnol'd method.} \label{Arnoldmethod}
In \cite{Arnold63a}, V. I. Arnol'd introduced a
method prove the persistence of
quasi-periodic solution
quite different from the method of proof
by Kolmogorov that we have discussed in
the previous Section \ref{Kolmogorovmethod}.
Rather than trying to perform a change of variables that produces
one torus, the method of \cite{Arnold63a} produces changes of
variables that reduce the system to approximately integrable in
a region of space. Hence, the method of \cite{Arnold63a}
produces all the tori at the same time.
The main complication that arises
with respect to the method of Kolmogorov
is that the intermediate steps require
to study transformations
that are defined in
rather
complicate regions. The fist transformation is
defined in a domain that
excludes the low order resonances.
(The places where $\partial_I H(I) \cdot k \ll 1 $ for
$|k|$ not big.)
In successive steps of the iterative procedure, one performs another
transformation that reduces the system much more closely to integrable,
but in a more complicated region since we need to take into
account more resonances.
At the end of the process one ends up with a transformation
defined on a Cantor family of invariant tori.
(A set which is locally diffeomorphic to the Cartesian
product of a Cantor set and a torus. Each of the torus
in a connected component of the set is invariant. )
An alternative way to describe the whole process is
to say that we have a smooth canonical
transformation defined in the whole space which reduces
the perturbed differential equation to integrable in a smaller set.
At intermediate steps of the iteration we just keep
estimates of how the system differs from integrable in
a smaller and smaller set with increasingly complicated
geometry. In the limit, we obtain control
on just a Cantor family of tori, on which the system
can be considered as integrable.
The basic strategy, which we will detail later,
has several advantages with respect to the
Kolmogorov one:
One of them is
that one obtains more information on the
way that the tori are organized.
For example, it follow rather naturally
that the tori constitute a family that is
differentiable in the sense of Whitney.
(This was observed in \cite{ChierchiaG82}
\cite{Gallavotti1}. Similar results can
were obtained by other methods in \cite{Poschel}.)
Another advantage is that if
we stop the process after a finite number of steps,
we may still have quite good information about the
system. For example, under the assumption that
$\frac{\partial^2 }{\partial I_i \partial I_j } H(I) $
is a positive definite matrix, in \cite{Nekhoroshev77}
(as a matter of fact, the assumption in \cite{Nekhoroshev77} is
sharper but more complicated to
state than the positive definiteness,
which is enough in many applications and which is the assumption
used in many more modern proofs.)
one can find the result that, denoting
by $\epsilon = \| R\|_\sigma $ in \eqref{hamiltonianperturbed}
we have,
for times $ t \le \exp(A \epsilon^{-a})$, {\bf all} the orbits of
the perturbed system \eqref{hamiltonianperturbed}
remain at a distance not more than $\epsilon^b$ of those of
the perturbed system.
The method of exclusion of parameters
near the resonances and continuing the transformation
in the rest of the space, has had many applications in other
KAM problems. For example, in the problem of changing
a system with quasi-periodic coefficients
into constant coefficients, usually
called the reducibility problem, most of
the papers (see specially the early ones \cite{DinaburgS75})
are quite influenced by the method. We refer to the lectures
of Prof. Eliasson in this meeting for an up to date
review of this problem. The strategy of
\cite{Arnold63a} was also employed in the first
proofs that started to study the problem of lack of
parameters and the related problem of studying systems which are
rather degenerate.
{From} the point of
view of the regularity assumptions needed
the main shortcoming of the method is that the
analytic part of the proof is based on truncating the Fourier series
of the perturbation
which produces bad results in finitely differentiable
systems. Even it it is not too difficult,
I know of no place in the literature where the
Arnol'd strategy is implemented for finitely differentiable
systems. (I wrote some very preliminary
notes on that for a graduate course.)
Another shortcoming arises from the
fact that one of the elements of the iterative
step is the domain of the definition on which the
changes of variables are defined.
Keeping track of this domain is much more complicated
than keeping track of the sizes of the functions.
Hence, the proofs are more complicated
and often one obtains worse estimates on the
sizes of perturbations allowed and other quantitative results.
(Nevertheless the method was used in the first proofs of
several sharp estimates such as \cite{Neishtadt81}
\cite{Wayne84}.)
I do not think that the method of \cite{Arnold63a} has been formalized
in such a way that it leads to an abstract implicit
function theorem in the style of Theorem \ref{Zehnder}
which takes care of the detailed estimates in applications or,
at least provided with a detailed strategy to carry them out.
Besides \cite{Arnold63a}, a very pleasant
and instructive modern
exposition of this method of proof is
\cite{Gallavotti1} (see also \cite{ChierchiaG82}.)
The Nekhoroshev theorem
proved by this method is nicely explained in
\cite{BenettinGG85} and a unified exposition of
Nekhoroshev and KAM theorems is
in \cite{DelshamsG96}. Other proofs of
Nekhoroshev theorems are covered in
\cite{Poschel93}, \cite{Lochak92}.
In somewhat more, but still insufficient, detail:
At the $n$th step of Arnold's method,
we keep track of:
\begin{itemize}
\item[1)]
An excluded set, on which we do not expect to define the
transformation.
\item[2)]
In the complement of the excluded set we have defined
a transformation $C^{(n}$ in such a way that
$$
H\circ C^{(n} = \bar{H}^{(n}(I) + R^{(n} \ ,
$$
\item[3)]
We keep track of
$\| \nabla H^{(0} - \nabla \bar{H}^{(n} \|_{\sigma_n}$
and $\| R^{(n} \|_{\sigma_n}$.
We assume by induction that
$\| \nabla H^{(0} - \nabla \bar{H}^{(n} \|_{\sigma_n}$
remains bounded and that $\| R^{(n} \|_{\sigma_n}$
is bounded by a superexponentially decreasing function.
(The $\| \cdot \|_{\sigma_n}$ norms will refer to
complex extensions of the excluded set, not a fixed set.)
\end{itemize}
Filling in more details about 1):
The excluded set consists of bands given by
\begin{equation}\label{excludeddefined}
\Bigl| \frac{\partial \bar{H}^{(i} }{\partial I} (I) \cdot k \Bigr|
\ge C_{n,i} |k|^{-\nu} \ , \quad
2^{i-1} < |k| \le 2^i \ .
\end{equation}
In particular, it is a set with piecewise smooth boundary
and the angles of the corners are bounded from below
by $C4^{-n}$ (where, $C$ again is a constant that depends on the
inductive assumptions).
This lower bound on the angles comes from the fact that
a bound of this sort is what one would get for planes
whose normals are integer vectors of total length $2^n$
and the fact that $\bar{H}^{(i}$ are
uniform diffeomorphisms and, therefore only change the angles
by a factor which remains uniformly bounded through all
the iteration.
We denote the excluded set by ${\mathcal E}_n$ and
\begin{equation}\nonumber
\begin{split}
{\mathcal D}_{n,\sigma} &:= \{ \, z \in \cee^d \times \cee^d/\zed^d \,
| \, d(z,{\mathcal E}_n) \le \sigma \, \} \\
\| f \|_{n,\sigma} &:= \sup_{z\in {\mathcal D}_{n,\sigma}} |f(z)| \ .
\end{split}
\end{equation}
Once we fix a sequence $\{\sigma_n\}$ (we will take
$\sigma_n = \sigma_0 (1 - \frac 12 \sum_{n=0}^\infty (\frac 13)^n )$),
we denote the norm $\| \cdot \|_{n,\sigma_n}$
by $\| \cdot \|_{n}$.
The main difference between these norms and the regular ones
is that, due to to the small angles, the Cauchy estimates are worse.
Nevertheless, given the lower bound on the angles,
they do not get too much worse:
$$
\| \nabla f \|_{\sigma_n e^{-\delta_n}} \le
C \left(\delta_n e^{-4^n}\right)^{-1} \| f \|_{\sigma_n} \ .
$$
To go from one step to the next, we exclude an slightly larger region
and define a new transformation
$C^{(n+1} = C^{(n} \circ \exp(\L_{G^{(n}})$
so that the new remainder is much smaller
(here, we will need to make a small modification
to our usual notion of smaller,
meaning quadratic times powers of the domain loss).
We see that
\begin{equation}\nonumber
\begin{split}
H^{(n} \circ \exp(\L_{G^{(n}}) =
& \,\,\bar{H}^{(n} + R^{(n} + \{ \bar{H}^{(n}(I), \, G^{(n} \} \\
& + \{ R^{(n}(I), \, G^{(n} \} + O ((G^{(n})^2)
\end{split}
\end{equation}
(a precise estimate for $O ((G^{(n})^2)$
appears in Lemma~\ref{leadingorder}).
A new idea of the method is to modify the prescription
of Newton method by restricting only to a finite number
of frequencies and include a truncation of the Fourier series
so that, at every stage, we only have to deal with a finite
(but growing) number of denominators.
The error incurred by the truncation can be estimated
if we increase the order of truncation at the right speed.
We write:
\begin{equation}\nonumber
\begin{split}
R^{(n\,[\le 2^n]} (I,\phi) &= \sum_{|k|\le 2^n}
\hat{R}^{(n}_k e^{2\pi ik \phi} \\
R^{(n\,[> 2^n]} (I,\phi) &= \sum_{|k|>2^n}
\hat{R}^{(n}_k e^{2\pi ik \phi} \\
\end{split}
\end{equation}
Hence, we solve:
\begin{equation}\label{linearizedarnold}
\{ \bar{H}^{(n} (I) , \, G^{(n} \} + R^{(n\,[\le 2^n]} (I,\phi)
= \Delta^{(n} (I) \ .
\end{equation}
The equation \eqref{linearizedarnold} can be solved by setting
\begin{equation}\label{arnoldsolution}
\hat{G}^{(n}_k (I) = {\hat{R}^{(n}_k (I)}/
({\frac{\partial \bar{H}^{(n}}{\partial I} \cdot k}) \ ,
\quad |k| \le 2^n \ .
\end{equation}
By the definition of the excluded set,
we can bound the denominators \eqref{arnoldsolution}
over the complement of the excluded set.
Notice also that we can bound
$$
\| R^{(n\,[> 2^n]} \|_{\sigma_n e^{-\delta_n}} \le
\| R^{(n} \|_{\sigma_n} e^{-\delta_n 2^n} \ .
$$
This allows us to define the generator of the transformation
that eliminates $R^{(n\,[\le 2^n]}$ (up to quadratic orders).
We have estimates
$$
\| G^{(n} \|_{\sigma_n e^{-\delta}} \le
C \delta^{-\tau} \| R^{(n} \|_{\sigma_n} \ ,
$$
where, as usual, $\tau$ is roughly $\nu$ plus something depending
on the dimension.
We use the letter $\tau$ to denote similarly constants that
depend only on the Diophantine exponent and the dimension.
To study the domain of $\exp(\L_{G^{(n}})$,
we note that if we set
$$
C_{n+1,\,i} = \delta^{-\tau} 2^n \| R^{(n} \|_{\sigma_n} + C_{n,\,i} \ ,
$$
we can define the transformation from the set
\begin{equation}\label{excludednew}
\Bigl| \frac{\partial \bar{H}^{(i}}{\partial I} \cdot k \Bigr| \ge
C_{n+1,\,i} |k|^{-\nu} \ , \quad
2^{i-1} < |k| \le 2^i \, \quad i = 1,2,\ldots,n
\end{equation}
to the set
$$
\Bigl| \frac{\partial \bar{H}^{(i}}{\partial I} \cdot k \Bigr| \ge
C_{n,\,i} |k|^{-\nu} \ , \quad
2^{i-1} < |k| \le 2^i \, \quad i = 1,2,\ldots,n \ .
$$
In that case, we have
$$
\bar{H}^{(n+1} (I) = \bar{H}^{(n} (I) + \bar{\Delta}^{(n+1} (I) \ ,
$$
from which it is clear that
$$
\| \bar{H}^{(n+1} \|_{\sigma_{n+1},\,{\mathcal D}_n} \le
\| \bar{H}^{(n} \|_{\sigma_n} + \| R^{(n} \|_{\sigma_n}
$$
and
$$
\| \nabla \bar{H}^{(n+1} - \nabla \bar{H}^{(n}
\|_{\sigma_{n+1},\,{\mathcal D}_n}
\le C 2^\tau \| R^{(n} \|_{\sigma_n} \ .
$$
Most importantly, we have:
\begin{equation}\label{importantrecursion}
\| R^{(n+1} \|_{\sigma_{n+1},\,{\mathcal D}_{n+1}}
\le C 2^{n\tau} \| R^{(n} \|_{\sigma_n, {\mathcal D}_{n}}
+ 2^{-\delta_n 2^n} \ .
\end{equation}
To define the next excluded set,
the only thing we have to do is to add to the excluded regions
corresponding to
$$
\Bigl| \frac{\partial \bar{H}^{(n+1}}{\partial I} \cdot k \Bigr| \ge
C |k|^{-\nu} \ , \quad
2^n < |k| \le 2^{n+1} \ .
$$
Of course, excluding more regions makes the suprema
in the left-hand side of \eqref{importantrecursion}
and all the other estimates even smaller.
The recursion \eqref{importantrecursion}
leads still to superexponential convergence
choosing $\delta_n = \delta_0 (2/3)^n$.
Establishing this was proposed in Exercise~\ref{improvements},
see \eqref{extraquadratic}.
Once we have the superexponential
convergence of the reminders, we obtain that the $C_{n,\,i}$'s
remain bounded and so does $\|\nabla \bar{H}^{(n}\|_{\sigma_n}$
Indeed,
$\|\nabla \bar{H}^{(0}- \nabla \bar{H}^{(n}\|_{\sigma_n}$
is small (arbitrarily small if we assume that
$\|R^{(0}\|_{\sigma_0}$ is sufficiently
small. Similarly, it is easy to
check that $\|(\nabla^2 \bar{H}^{(n})^{-1}\|_{\sigma_n}$
remains bounded and that the bound is close to
the one for $\|(\nabla^2 \bar{H}^{(0})^{-1}\|_{\sigma_n}$
if $\|R^{(0}\|_{\sigma_0}$ is sufficiently small.
Hence, under the assumption that $\|R^{(0}\|_{\sigma_0}$
is sufficiently small, we can verify
the inductive assumption
on $\|(\nabla^2 \bar{H}^{(n})^{-1}\|_{\sigma_n}$.
The passage to the limit in this procedure
is somewhat subtle.
In the original coordinates, we have to study
the sets
$(C^{(n})^{-1} {\mathcal E}^n$.
These sets will be dense.
By increasing slightly the excluded sets
at each stage so that we exclude also the mismatches
of the domain,
we can arrange that $(C^{(n})^{-1} {\mathcal E}^n$
are increasing.
(note that this extra exclusion will be decreasing
superexponentially since the transformations
that we need to carry out in each step are
decreasing superexponentially)
Hence,
$(C^{(n})^{-1} (\TT^d\times \RR^d - {\mathcal E}^n) $
is a decreasing sequence of compact sets.
On the other hand, their measure remains bounded away from zero
as follows from the fact that
$\|(\nabla^2 \bar{H}^{(n})^{-1}\|_{\sigma_n}$
remains uniformly bounded
so that we can use the same arguments as in Section~\ref{Diophantine}).
It is slightly more subtle, but we can also estimate the derivatives
of the transformations $C^{(n}$ to show that the derivatives
remain bounded
(it follows by an argument very similar to that used
in the proof of Theorem \ref{Siegel2} part v) )
This shows that the sets
$(C^{(n})^{-1} (\TT^d\times \RR^d - {\mathcal E}^n) $
get closer and closer to being invariant.
The limiting set will be invariant.
If one keeps track of all the derivatives in the closed sets,
one can show that the limiting transformation $C^{(\infty}$
is differentiable in the sense of Whitney
(see \cite{ChierchiaG82} or \cite{Gallavotti83}).
An interesting remark \cite{Valdinoci98}
is that one can use the fact that the gaps between the sets
are much larger than the corrections to show directly
the Whitney extension theorem.
This remark could be important when studying infinite
dimensional systems (e.g.\ PDE's).
In infinite dimensions, the Whitney extension theorem
is not a available, but the method of \cite{Valdinoci98}
could still work to produce tori that lie in a smooth family.
For more details of this method of proof
we refer to the original paper \cite{Arnold63a},
and the more expository paper \cite{Arnold63b}, which also contains
applications to celestial mechanics.
An early development of the method with several improvements is
\cite{Svanidze80}.
More modern expositions (including the Whitney differentiability)
of Arnol'd method are \cite{ChierchiaG82} and \cite{Gallavotti83}.
An exposition of the Arnol'd method that,
at the same time proves Nekhoroshev's theorem
and clarifies the geometry of the domains, is \cite{DelshamsG96}.
The method also lies at the heart of several other papers.
One paper that incorporates the exclusion of parameters
but is free of many geometric complications is
\cite{DinaburgS75}.
This paper also shows that the method can allow
some frequencies that are not Diophantine
(they allow
$|\omega\cdot k|^{-1} \ge \exp{\frac{A|k|}{\log|k|^{1+\ep}}}$).
The method of transformations and exclusion
of parameters is the basis of many modern developments
in KAM theory related to lower dimensional tori,
e.g.\ \cite{Eliasson88}, \cite{JorbaS92},
\cite{JorbaV97b}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Lagrangian proof.} \label{Lagrangianmethod}
In this section, we study a proof of
of the Theorem
\ref{KAMmaps} which has a different
flavor from the proofs already presented.
We will present the proof
only in the case $d=1$ and only for the
particular case of the map
given in \eqref{standardmap}. Similar proofs
in any dimension and for more general maps
are in the literature and we refer to that.
The proof differs substantially from the previous proofs
of Theorem \ref{KAMflows} in
that it does not use compositions.
Of course, the proof we presented of
Theorem \ref{Siegel1} does
not require compositions either, even if
the proofs we have presented so far for
Theorems \ref{KAMflows} do rely on transformation theory/
More interesting is that it is based on Lagrangian formalism.
(That is, on second order equations rather than in systems
of first order equations. The structure that is
used is the fact that the equations solve a
Lagrange variational principle, not that they come from
a Hamiltonian formalism.)
The proof we present is based on unpublished notes of
J. Moser for a course he gave in Z\"urich.
A generalization of these results is included in the paper
\cite{SalamonZ89}. We follow very closely the presentation
in one of the chapters of \cite{Rana87}
(which in turn followed the presentation of the
Moser's course.)
In \cite{Rana87}, one can also find
the implementation of computer assisted proofs
based on this method. In particular the result that the map
given in
\eqref{standardmap} for $V(x) = \ep \frac{1}{2 \pi} \sin( 2 \pi x)$
has an invariant circle with golden mean rotation
for $\ep = .93$
(this was later improved to $\ep = 0.935$ ). This is very close
to the values for which \cite{Jungreis91} showed that there
can be no invariant circle.
We discuss some of these issues in
Section \ref{compassisted}.
\footnote{The conjecture in
\cite{Greene79}, given a theoretical -- but not yet rigorous -- basis in
\cite{McKay82} is that there are smooth invariant circles
with rotation golden mean
when $\ep < \ep^*$ and not when $\ep > \ep*$. For
$\ep = \ep^*$ there is an invariant circle which is not very
differentiable. It is believed that $\ep^* \approx .971635$.
Of course, in other families it could -- probably does -- happen
that the set of parameters for which one can find
an smooth invariant circle is a more complicated set, perhaps
with infinitely many components.
}
The Lagrangian formalism for KAM theory has several other
applications. For example, many elliptic PDE's have
a very natural Lagrangian formalism but not a
simple Hamiltonian one. (Note that in this case,
the independent variable is multidimensional, while
in Mechanics, the independent variable is the time, which is
one-dimensional)
There is no easy canonical transformation theory for
elliptic PDE's.
We will try to find solutions to
\eqref{linstedtlagrange}
which read
\begin{equation}
\label{linstedtlagrange2}
\ell_\ep (\theta+\omega) + \ell_\ep(\theta-\omega) - 2\ell_\ep (\theta)
= - \ep V' (\theta+\ell_\ep(\theta))\ .
\end{equation}
We refer to Section \ref{linstedt} for the interpretation of
this equation as a parameterization of a set in
which the motion is quasiperiodic of frequency $\omega$.
Somewhat informally,
what we will do is to show that
there is a procedure that, given an
an approximate solution
of \eqref{linstedtlagrange2}
(which is not too badly behaved)
we can produce another function that solves
the equation even more approximately.
Then, we will have to show that the whole process can be
iterated indefinitely and that it converges to a solution.
Of course, making precise the notion of close, will involve
introducing
analytic norms. The statement that the result of the algorithm is
closer to being a solution
will mean to prove that in an slightly smaller domain, we will have
the usual bounds which are quadratic in the previous error and
have powers of the loss of analyticity.
The fact that the iterative step can be performed
and that it will lead to the desired improvement will require that
certain expressions are not too large.
(This is what we alluded to when mentioning
that the solution is well behaved.) Of course, we will need to
check that, if the initial error is small enough, the quadratic
convergence that ensues, allows us to recover the inductive hypothesis
indefinitely.
The theorem whose proof we will sketch is:
\begin{theorem}\label{KAMlagrangian}
Let $\ell_0: \torus^1 \to \torus^1 \times \real^1$
be such that $\| \ell_0\|_\sigma < \infty$.
Assume that
\begin{equation} \label{1apm}
\begin{split}
&\| \ell^\prime_0 + 1 \|_\sigma \leq M_+, \\
&\| (\ell^\prime_0 +1)^{-1} \|_\sigma \leq M_-
\end{split}
\end{equation}
\begin{equation} \label{1b}
\frac{1}{2}
\| V^{\prime\prime} \|_{\sigma e^{( M_+ + \frac{3} {M_-})} } \leq D
\end{equation}
and
\begin{equation}
\| \ell_0 (x + \omega) - 2 \ell_0 (x) + \ell_0 (x - \omega) - V'(\ell_0 (x) + x)
\|_{\sigma} \leq \epsilon
\end{equation}
where $M_+, M_-, D$, and $\epsilon$ are finite positive constants.
Let $\Gamma(M_+,M_-, D, K, \nu)$ be a
function which will be made
rather explicit during the proof, where $K$ and $\nu$
are the constant and the exponent appearing in the
Diophantine properties of $\omega$.
If
$$
\epsilon \le \Gamma(M_+,M_-, D, K, \nu)
$$
then,
there is a periodic function $\ell, \ \ \ell (x+1) = \ell (x)$, \ solving
\eqref{linstedtlagrange2}.
Moreover
$$
\| \ell - \ell_0 \|_{\sigma_0/2}\leq C \epsilon
$$
where $C$ is a constant that depends on $M_+,M_-,D,K,\nu$.
\end{theorem}
The proof will be done using a quasi-Newton method.
The method will be rather similar to the proof of
Theorem \ref{Siegel1}.
We try to solve the infinitesimal equation
suggested by the Newton method.
This will lead to an equation which is not immediately
solvable with the method of Section~\ref{linear_estimates}.
Nevertheless, by manipulating the equation with the remainder,
we will arrive at a factorization of the equation that
will be solvable by applying repeatedly the
theory for the equation \eqref{diff}.
Denote
\begin{equation} \label{lagrangianfunctional}
\Tau(\ell)(x) \equiv
\ell (x + \omega) - 2 \ell (x) + \ell (x - \omega) - V'(\ell(x) + x)
\end{equation}
We assume that we are given an approximate solution $\ell$ such that
\begin{equation}\label{lagrangianremainder}
\Tau(\ell) = R
\end{equation}
where $R$ is small.
The prescription of the Newton method would be to
find a $\Delta$ periodic solving
\begin{equation}\label{newtonlagrangian}
\Delta(x + \omega)
+ \Delta(x - \omega) - (2 + V''(\ell(x) + x) ) \Delta(x) = -R(x)
\end{equation}
This equation \eqref{newtonlagrangian}
is not readily solvable in terms of Fourier coefficients
(as indicated in Exercise \ref{secondordereq})
since the term $( 2 + V''(\ell(x) + x)) \Delta(x) $ is not diagonal
in Fourier coefficients. Our next task is to manipulate the
equation so that it becomes solvable using the
Fourier methods. The manipulations that will follow,
even if rather straightforward and indeed convenient
for numerical work
will perhaps look mysterious, but in later sections
we will argue that the success is due to natural
geometric reasons.
If we take derivatives with respect to $x$ of
\eqref{lagrangianremainder}, we obtain,
denoting by $T_\omega(x) = x + \omega$
and by $g'(x) = \ell'(x) + 1$
\begin{equation}\label{derivativeremainder2}
g'\circ T_\omega + g'\circ T_{-\omega} - (2 + V''\circ g) g' = R'
\end{equation}
Substituting the expression for $2 + V''\circ g$
from \eqref{derivativeremainder2} into \eqref{newtonlagrangian},
we obtain
\begin{equation}\label{step3}
g' \Delta\circ T_\omega
+ g' \Delta \circ T_{-\omega} -
[-R' + g'\circ T_\omega + g'\circ T_{-\omega}] \Delta = - g' R
\end{equation}
Ignoring the term
$R^\prime \Delta$, which is quadratic in the error,
yields the system of equations:
\begin{equation} \label{substitution1}
\left( \frac{\Delta} {\ell^\prime + 1} \right) \circ T_{-\omega} -
\left(
\frac{\Delta }{\ell^\prime+1} \right) =
\frac{W}{(\ell^\prime +1)
(\ell^\prime_0 +1) \circ T_{-\omega}}
\end{equation}
with:
\begin{equation} \label{substitution2}
W \circ T_\omega - W = (\ell^\prime +1) R
\end{equation}
The above system of equations consists of equations of the form
\eqref{diff} which can be studied using
Lemma~\ref{linearestimates}. We first solve
\eqref{substitution2} for $W$ and we take the solution
and substitute it in the \RHS of
\eqref{substitution1}. We then solve
\eqref{substitution1} for $\frac{\Delta }{\ell^\prime+1}$,
out of which $\Delta$ is obtained just multiplying by ${\ell^\prime+1}$.
Of course, in order to carry out the above plan, we need to
check that the equations we plan to solve are indeed
solvable (i.e. that their \RHS has average zero).
Later, we will have to worry about obtaining estimates of
the solution thus obtained.
The fact that \eqref{substitution2} can be solved
is a calculation which we have done in
Section~\ref{linstedt} when we wanted to
show the solvability of
equation \eqref{norder}. Once we have that the
equation \eqref{substitution2} is solvable up to an additive constant,
we can determine the additive constant in $W$ in such
a way that the \RHS of \eqref{substitution1}
has average zero.
(Note that adding a constant $\bar W$ to $W$ changes the average of
the \RHS of \eqref{substitution1}
by $\bar W \cdot \int
( (\ell^\prime +1)
(\ell^\prime +1) \circ T_{-\omega})^{-1} $.
The integral is not zero, since we assumed by induction that
the denominator in the integrand is bounded away from zero and
hence, it is positive.
Indeed, we can have a bound for it under the assumptions given
in our inductive hypothesis.
The fact that this procedure works can be
shown using the familiar method.
Adding and subtracting, we show that given some inductive
assumptions on bounds on $||\ell' + 1||_\sigma$,
$||(\ell' + 1)^{-1}||_\sigma$, the above procedure leads to a
new quadratic remainder. That is, as usual, we have
$$
||R^{n+1} ||_{\sigma_n - \delta_n } \le
||R^{n} ||_{\sigma_n}^2 C K^2 \delta_n^{-\tau}
$$
The bounds we assumed on the derivatives deteriorate slightly, but again
the quadratic convergence ensures that they remain bounded during the
iteration.
\begin{remark}
The remarkable cancellations
(see \eqref{derivativeremainder2}, \eqref{newtonlagrangian})
between the derivative of the remainder and the linearized
equation which allowed us to obtain a quadratically convergent
method solving only the linear equation (and performing
easy multiplication and divisions by known functions)
are not
a coincidence. In \cite{SalamonZ89} one can find how
they work for twist maps if we uses the equations given by the
generating functions, linking them to a Lagrangian formalism.
Indeed there are deeper reasons. For example, the cancellations
apply to partial differential equations. See \cite{Kozlov83},
\cite{Moser88}.
\end{remark}
\begin{remark}
If one is interested in obtaining the existence of
invariant circles for numerical values that are as close
to the optimal value as possible, one should be
prepared to cope with the difficulty of having the quality of
the solution get worse and worse.
Indeed, the domains of analyticity shrink and function
becomes more and more close to having zeros.
Indeed, there are precise predictions -- not rigorous
but supported by numerical evidence that at the breakdown
of the invariant circle for
the map given in \eqref{standardmap},
all the difficulties happen at the same time
and indeed all the quantities that need to be estimated blow
up as powers of the distance of the parameter to the critical
one.
See \cite{BerrettiCCF92} for numerical
results and \cite{Llave92} for a non-rigorous explanation
and precise conjectures.
\end{remark}
\subsection{Proof without changes of variables.} \label{nochangemethod}
In this section, we will present another proof
of theorem \ref{KAMmaps}
This proof is based on \cite{JorbaLV00}.
A version of the method for
lower dimensional tori was presented in
\cite{JorbaLZ00}.
The proof actually proves something more general since
the main result does not require that the map is
exact. Of course, without
assuming exactness, we cannot expect to have invariant tori
as was shown in the examples.
The conclusion of the main theorem is that
for symplectic maps that satisfy
all the other assumptions of
Theorem \ref{KAMmaps}
there is a torus that gets translated rigidly
in the direction of the actions.
The points of the torus are, roughly, rotated.
This is a generalization to higher dimensions of
the translated curve theorem of
\cite{Russmann76}.
If we assume that the map is exact, it will be
very easy to show that the translation has to vanish
and that the torus is indeed invariant and that the motion on
it is conjugated to a rigid rotation.
We will consider the symplectic manifold
$\torus^d \times \real^d$ endowed with the standard symplectic
structure.
We will consider a map $F:\torus^d \times \real^d \to \torus^d \times \real^d$
which is symplectic (not necessarily exact)
analytic (and other conditions somewhat
weaker than those of Theorem \ref{KAMmaps}
which we will formulate when the
heuristic discussion motivates them).
We fix $\omega$ Diophantine
and seek a mapping
$K: \torus^d \rightarrow \torus^d \times \real^d$
and a vector $a \in \real^d$
in such a way
that
\begin{equation} \label{parametrization}
F\circ K(\theta) = K( \theta + \omega) + (0, a)
\end{equation}
Note that, of course, the equation
\eqref{parametrization} expresses that the image of the
torus is translated in the direction of the action by
a rigid displacement $a$.
In the case that $F$ is an integrable map, all the tori
given by a parameterization
\begin{equation} \label{integrableexample}
K_0(\theta) = (\theta, I_0) \
\end{equation}
are ``vertically'' translated.
So, we expect that the functions we will have
to consider will be close to that.
Later, we will show that if $F$ is exact (and there are other
conditions), then, $a$ should vanish. This is very similar to
the line of argument in \cite{Russmann76}).
When $d = 1$, it can be seen that the zero flux condition indeed
implies that $a = 0$.
We note that, even if the proof does not use the exactness of the
symplectic structure, it uses the symplectic structure.
Under appropriate redefinition of translation, one can have similar
theorems in other symplectic manifolds.
We will sketch the proof of the following theorem:
\begin{theorem}\label{translatedcurve}
Assume that $F$ is an analytic symplectic map of
$\torus^d \times \real^d$ endowed with the
canonical symplectic structure and that $\omega$ is a Diophantine
number.
Assume that $F$ is close to an integrable map and that it satisfies
the hypothesis of non-degeneracy of Theorem \ref{KAMmaps}.
Assume that we can find an approximate $(K, a)$
solution of \eqref{parametrization}.
If the residual of \eqref{parametrization} is small enough
(depending on properties of $F$ and of $\|K - K_0\|$ )
where $K_0$ is the solution in \eqref{integrableexample}),
then we can find
an exact solution of \eqref{parametrization}.
In particular, if we take as approximate solution $K_0$
as in \eqref{integrableexample}
the hypothesis are satisfied when $F$ is sufficiently close
to an integrable map.
\end{theorem}
The fact that $F$ is close to integrable is not
really necessary as it will transpire from the proof.
At this stage it is only introduced to
avoid using a more complicated notion of
non-degeneracy than that used in
Theorem \ref{KAMmaps}. As the proof in
Section \ref{Lagrangianmethod} it can apply to
all the maps of the form \eqref{standardmap}.
Even that can be generalized by formulating
appropriately the degeneracy.
See \cite{JorbaLV00}.
A very simple calculation
(a more general version appears in \cite{JorbaLZ00})
shows that if
we have an exact system, then the translation $a$
for an true solution has to be zero.
\begin{proposition}\label{vanishing} If the $K$ solving \eqref{parametrization}
is close to $K_0$ in an analytic norm and
$F$ is exact, then $a = 0$.
\end{proposition}
Before starting the discussion of the Theorem~\ref{translatedcurve},
we discuss the proof of Proposition~\ref{vanishing}.
\begin{proof}
Let $\alpha$ be the symplectic potential form
$\alpha= \sum_i I_i \ d \phi_i$. Assume also
that $F^* \alpha= \alpha + d S$.
We consider the loops in the $i$th angle coordinate given by
$$
L_{\theta_1, \cdots, \theta_{i -1}, \theta_{i+1} ,\cdots,\theta_d} (\theta) :=
K(\theta_1, \cdots, \theta_{i -1}, \theta, \theta_{i+1} , \cdots ,\theta_d ),
$$
where
$\theta_1, \cdots, \theta_{i -1}, \theta_{i+1} , \cdots ,\theta_d
\in \torus$.
Because of \eqref{parametrization} and the exactness of the map,
we have
$$
\int_
{L_{\theta_1, \cdots, \theta_{i -1}, \theta_{i+1} , \cdots ,\theta_d} }
\alpha =
a_i +
\int_
{L_{\theta_1 +\omega_1, \cdots, \theta_{i -1} + \omega_{i-1},
\theta_{i+1}+\omega_{i+1} , \cdots ,\theta_d +\omega_d}} \alpha
$$
And, integrating over the variables
$
\theta_1, \cdots, \theta_{i -1}, \theta_{i+1} , \cdots\theta_d
$, we obtain
$$
a_i = 0
$$
\end{proof}
\begin{remark}
This Theorem (and the Proposition)
are much weaker than what can be proved by the method.
For example, the hypothesis that the system is close to
integrable can be replaced by several quantitative statements
about the approximate solution. This is quite important
for several applications. We note that approximate
embeddings can be obtained with the computer. One can try
to solve a discretized Fourier series or, a big advantage for
the present method, just compute orbits and compute the
Fourier transform.
This improvement is discussed in more detail
in Remark~\ref{rem:alternative}.
Once this improvement is in place, it should be apparent that the
way that the torus is embedded does not play any role.
We do not need that the system is close to integrable
or that the tori are close to integrable.
(This is the main difference with the exposition of
\cite{Bost86}.)
In particular,
we can justify the existence of tori which have
different topology than
the tori of the unperturbed system.
(Recently there has been some interest in these
secondary tori since there are numerical
experiments that suggest that secondary tori are very important for the
statistical properties of coupled systems \cite{HaroL00}.)
Also, an important advantage of the method is that it allows one
to deal will more degenerate situations than the twist mapping.
Indeed, one can use it to deal with non-twist maps and with
even more degenerate situations.
We refer the reader to \cite{JorbaLV00} for these precisions as well
as for more details about the proof.
We also refer to \cite{JorbaLZ00} for another application
of similar techniques to discuss lower dimensional tori.
\end{remark}
Now, we start describing the main ideas of the proof.
Again, we refer to \cite{JorbaLV00} for more details.
The method of proof will be an iterative procedure
in which we start from \eqref{parametrization} being satisfied
with a certain error and return a solution that satisfies the equation
with an smaller error. Of course, as usual in this theory, what
we mean by smaller error is that the size of the new error
will be bounded (in a smaller domain than the original
one) by the square of the size of the original error
times a factor that is the domain loss parameter to
a negative power. Of course, by now the convergence
of the procedure should be well understood. Actually, since
we do not need to make changes of variables and we do not
need to keep track of much the geometric structures,
the inductive hypothesis will be very mild.
We will begin with a heuristic discussion.
If we start with an approximate solution of
\eqref{parametrization} that is:
\begin{equation}\label{approximatepar}
F\circ K(\theta) - K(\theta + \omega) - (0, a) = R(\theta)
\end{equation}
where $R$ is small in some appropriate norm that we will make
precise later.
The Newton method prescription, would be to change
$K$ into $K + \Delta$, $a$ into $a + \alpha$ in such a way
that
\begin{equation}\label{newtonpar}
DF\circ K(\theta)\ \Delta(\theta) - \Delta(\theta + \omega)
- (0,\alpha) = - R(\theta) \ .
\end{equation}
Unfortunately, this equation is not readily solvable
by easy methods such as comparing Fourier coefficients
since it involves the non-constant coefficient
factor $DF\circ K(\theta)$.
Hence, we try to compare it
with the equation obtained taking derivatives
of \eqref{approximatepar}:
\begin{equation} \label{approximateparder}
DF\circ K(\theta) \partial_\theta K(\theta) -
\partial_\theta K(\theta + \omega ) = \partial_\theta R(\theta) .
\end{equation}
At this point, we are going to introduce some notation
(which is not completely necessary but which will make the
geometry more concrete). We define
$D(\theta) := DF \circ K(\theta)$
and let
$K_1(\theta)$
an orthogonal basis
for $\partial_\theta K(\theta)$.
The previous equation \eqref{approximateparder}
reads $D(\theta) K_1(\theta) - K_1(\theta + \omega) = R_1(\theta)$.
As usual, we define the matrix
$$ J =
\left(
\begin{array}{cc}
0 & \Id_d \\
-\Id_d & 0
\end{array}
\right) \ ,
$$
which is the representation in
coordinates of the symplectic form.
We define then the symplectic matrices.
\begin{equation}\label{Mmatrix}
M(\theta) = [ K_1(\theta), J K_1(\theta)]
\end{equation}
Notice that from the fact that
$K_1$ is almost invariant under $DF(\theta)$
we obtain that:
\begin{equation} \label{invariantform}
DF(\theta) M(\theta) = M(\theta + \omega)
\left(
\begin{array}{cc}
\Id & A(\theta) \\
0 & B(\theta)
\end{array}
\right) \, + O( R )
\end{equation}
We will introduce the assumption that
$M(\theta)$ is invertible for all
$\theta$.
This is reasonable assumption in view of
the fact that, for integrable systems
\footnote{
Indeed, this is the only reason why we assumed
that $F$ was close to integrable. If we
formulate the theorem assuming that
$M$ is invertible, we could have eliminated the
assumption of close to integrability.
Later, we will need to formulate the non-degeneracy
assumption using this matrix $M$.
}
This is explained in more detail in
Remark~\ref{rem:alternative} and
in the references quoted there.
using \eqref{integrableexample}
we have:
$$
M(\theta) =
\left(
\begin{array}{cc}
\Id_d &0 \\
0 & \Id_d
\end{array}
\right) \ .
$$
Recall also that
the assumption that the map $F$
preserves the symplectic form
is equivalent to
\begin{equation} \label{preservation}
J DF(x) = \left [ DF(x)^t \right]^{-1} J \ .
\end{equation}
This gives that $B(\theta) = \Id_d$.
We note that in the integrable case, the matrix
$A(\theta)$ will be a constant $d \times d$ matrix $A$ and,
the twist condition implies that $A$ is invertible.
Hence, in the proof of the theorem, we will assume that
$A(\theta)$ is not very far
from a constant, invertible matrix
in the sense that $\bar A(\theta) $ is an invertible matrix.
Indeed, this is the only non-degeneracy condition that we will
assume.
We call attention to the fact that the non-degeneracy assumption
only amounts to the invertibility of $M$ and the
fact that $\bar A(\theta) $ is invertible.
These assumptions could be checked a posteriori on
a numerically computed solution or on an approximate
solution produced by any other means. Other than that,
we do not need any property of the map $F$.
See Remark \ref{rem:alternative} and the references quoted there
for an explanation of this alternative approach.
\begin{remark}
It is an easy exercise to show that, under Diophantine conditions
we can reduce the block $A(\theta)$ to a constant, so that the matrix
is is indeed reducible, Nevertheless, for the applications that
we have in mind, this does not help. Indeed, by doing it, we incur
in extra small denominator estimates, which can worsen the result.
\end{remark}
\begin{remark}
A more geometric interpretation of the previous
calculations is to say that
$DF \circ K(\theta)$ is a reducible matrix
whenever $K$ is a parameterization of
an invariant torus by a rigid rotation.
We want to give a geometric argument that shows
that the linearization of the equations around
an invariant torus is reducible.
The argument will show that for an approximate
solution, the equation will be approximately
reducible and, hence that one can start
an iterative procedure in which in the iterative
step we improve the solution of the main equation and
its reducibility.
That is, our goal find a system of coordinates on the tangent
of the torus so that the matrix representing
$DF \circ K(\theta)$ has constant coefficients.
Since the vectors along the direction of $\theta$ are
moved just by a rotation in the torus, this is an invariant
field that can be lifted to the space by the embedding.
By the preservation of the symplectic structure,
we also have that the plane spanned by the the
vector and its symplectic conjugate is
also preserved.
We can see that in the plane spanned
by a vector and its symplectic the matrix
has to be upper diagonal (one vector is preserved.)
The dilation along the symplectic conjugate has to be the
inverse of the dilation along the preserved direction
due to the requirement that the two-area in the plane is
preserved. This gives us the diagonal blocks of the
matrix. The upper diagonal does not bother.
This system of coordinates
provides with a system in which the derivative is
upper triangular.
Once that we have that the diagonal blocks are constant,
then it is easy to see that the linearized equation can be
solved by using equations of the form \eqref{diff}.
The above geometric interpretation makes it clear
that we do not need the symplectic form
to be constant. Moreover, it is clear that it
does not require that the symplectic form has
action-angle variables and that it can
accommodate certain singularities.
\end{remark}
The algorithm is now very easy.
If we write $\Delta(\theta) = M(\theta) w(\theta) $
and substitute in \eqref{newtonpar}
we obtain
\begin{equation}
D(\theta) M(\theta) w(\theta) - M(\theta + \omega) w(\theta + \omega)
- (0,\alpha) = - R(\theta)
\end{equation}
which using \eqref{Mmatrix}
becomes:
\begin{equation} \label{transformed}
M(\theta + \omega) \left[
\left(
\begin{array}{cc}
\Id_d &A(\theta)\\
0 & \Id_d
\end{array}
\right) w(\theta) - w (\theta + \omega) \right]
-(0,\alpha) = - R(\theta) - N(\theta) w(\theta) \ .
\end{equation}
Therefore, ignoring the
last term of the \RHS of \eqref{transformed}, which
is quadratic, we are lead to the
study of the equation:
\begin{equation}\label{tosolvepar}
\left(
\begin{array}{cc}
\Id_d &A(\theta)\\
0 & \Id_d
\end{array}
\right) \left[ w(\theta) - w (\theta + \omega) \right]
= - M(\theta + \omega)^{-1} \left[ R(\theta) - (0,\alpha) \right]
\end{equation}
We claim that this equation for $w$, $\alpha$
can be studied using the methods that we have
developed in Section \ref{linear_estimates}.
This will constitute our iterative step.
Of course, after this heuristic derivation, we will
need to go back and justify the estimates of the step
and show that it can be iterated.
This, even if being the essential part of the proof,
we hope will bring no surprises anymore for the reader.
If we write \eqref{tosolvepar} in components,
denoting the components of
$w(\theta) = (w_\phi(\theta), w_I(\theta) ) $
and by $\Pi_\phi$,$\Pi_I$ the projections over the
components,
we have:
\begin{equation}\label{tosolveparcomp}
\begin{split}
w_\phi(\theta) + A(\theta) w_I(\theta) -
w_\phi(\theta + \omega) & =
- \Pi_\phi M(\theta + \omega)^{-1}
\left[ R(\theta) - (0,\alpha) \right] ) \\
w_I (\theta) - w_I (\theta + \omega) & =
- \Pi_I M\left(\theta + \omega)^{-1}
\left[ R(\theta) - (0,\alpha) \right]
\right)
\end{split}
\end{equation}
If we look at the second equation in
\eqref{tosolveparcomp}
(recall that it is an equation for $w_\phi$ and $\alpha$)
we see that it is an equation of the form
\eqref{diff} which we have already studied.
We chose $\alpha$ in such a way that the \RHS
has average $0$. (This can be done if
$M$ is close to the identity, but otherwise
it can be made into an assumption to be checked
a posteriori on the approximate solution.)
Note that we have bounds
\begin{equation}\label{alphabounds}
|\alpha| \le C\|R\|_\sigma \|M^{-1}\|_\sigma \|
(M^{-1} - \Id)^{-1}\|_\sigma\end{equation}
If we assume for convenience
(somewhat sharper assumptions could also
work, see Remark \ref{rem:alternative}) that the
factors in the \RHS of \eqref{alphabounds}
satisfy:
\begin{equation}\label{inductive1par}
\| M^{-1}\|_\sigma \| (M^{-1} - \Id)^{-1}\|_\sigma \le 333 \ .
\end{equation}
Then we can apply Lemma \ref{linearestimates} to
obtain $w_I$ up to a constant,
which we will determine in the next equation.
We have, denoting by $\tilde w_I$ the solution with
zero average:
\begin{equation} \label{wIestimates}
\| \tilde w_I\|_{\sigma -\delta} \le C \delta^{-\nu} \|R\|_\sigma \ .
\end{equation}
If we look at first equation of
\eqref{tosolveparcomp}
we see that, at this stage of the argument is
an equation only for $w_\phi$ and the average of
of $w_I$.
Hence, we write it as
\begin{equation}\label{firsteqtransformed}
w_\phi(\theta) - w_\phi(\theta + \omega) =
- \Pi_\phi M(\theta + \omega)^{-1} \left[ R(\theta) - (0,\alpha) \right] )
- A(\theta) w_I(\theta) \ .
\end{equation}
If we assume that
\begin{equation} \label{inductive2par}
\| (\bar A)^{-1} \| \le 333 \ ,
\end{equation}
we can determine
$\bar w_I$ so that the terms in the \RHS of
\eqref{firsteqtransformed} have average
$0$. We have:
\[
| \bar w_I | \le C\|R\|_\sigma
\]
We will furthermore assume that
\begin{equation} \label{inductive3par}
\| A \|_\sigma \le C
\end{equation}
Hence, we can apply Lemma \ref{linearestimates}
and obtain a $w_\phi$ with zero average
which satisfies:
\begin{equation}\label{wphiestimates}
\|w_\phi \|_{\sigma - 2\delta } \le C\delta^{-2\nu} \|R\|_\sigma \ .
\end{equation}
Note that the power of $\delta$ in this case is twice as high
as that in the previous one since the \RHS of
\eqref{firsteqtransformed} involves the solution of the previous one.
>From \eqref{wIestimates}\eqref{wphiestimates}, using the inductive
assumptions on the size of $M$, we obtain
$$
\|\Delta\|_{\sigma - 2 \delta } \le C \delta^{-\tau} \|R\|_\sigma .
$$
>From this, the rest of the proof of the translated
tori theorem is very similar to the
previous proofs, in particular to the proof of
Theorem \ref{Siegel1}.
Under the assumption that
\begin{equation}\label{inductive0par}
\|K \|_\sigma + \| \Delta\|_{\sigma - 4 \delta } \le \Sigma - \delta
\end{equation}
where $\Sigma$ denotes the size of the domain of analyticity
of $F$, we can define the composition $F\circ( K + \Delta)$
and indeed the range of $K +\Delta$ is at least a distance
$\delta$ from the boundary of the domain of definition
of $F$.
Note that adding and subtracting and using Taylor's theorem
to control the terms neglected to derive \eqref{newtonpar},
(and Cauchy bounds to control the size or the derivatives
involved)
we get:
\begin{equation}
\|\tilde R\|_{\sigma -4 \delta} \le C \delta^{-\tau'} \| R\|_\sigma^2 \ .
\end{equation}
>From this, we can conclude as in the previous cases that if
the original remainder is small, then the iteration can be carried
out an arbitrarily large
number of times, moreover, the final remainder in its domain
of definition keeps decreasing.
Note also that this proof -- in contrast with those based on
composition -- does not require any subtle inductive hypothesis
to ensure that the domains of the composition match.
These assumptions, that we had to consider in the proofs based on
composition are subtle because they
require that the errors decrease faster than
the analyticity losses.
In this case, the only assumptions that we have to check
are
\eqref{inductive0par},
\eqref{inductive1par},
\eqref{inductive2par},
\eqref{inductive3par}.
We can see that if we start with an small enough residual,
the iterative procedure does not change $A$ or $M$ much,
so that using in the step bounds which are twice the ones at the
start, the estimates of the step remain valid if
the original error is small enough.
\begin{remark}
We emphasize that the only thing that we need to get the proof
started is an approximate solution of the functional
equation.
This can be obtained in a variety of ways. For
example if the system was close to integrable,
one could take as an initial guess the
parameterization of the integrable system.
Other choices are possible. One could
use a few steps of the Lindstedt series.
In such a case, the proof will establish that the
Lindstedt series is asymptotic.
More audaciously, one could use
the results of a non-rigorous, numerical
algorithm. Provided that one can verify
rigorously that one has an approximate solution,
one then obtains a rigorous proof of the existence of
these circles. These issues will be explored in
more detail in Section \ref{compassisted}.
We also note that the present proof does not require
much from the function except that it gives
a parameterization of an invariant torus.
In particular, it can apply to tori of topological
types not present in the original system.
\end{remark}
\begin{remark}
Another proof without changes of variables
can be found in \cite{Bost86}
which is based in unpublished work of
M. Herman. This proof contains
also a translated curve theorem.
The main difference with the proof presented here is
that that method parameterizes the curves by the graph of
a function. When studying tori that are not graphs,
it requires that one performs a preliminary change of
variables.
\end{remark}
\begin{remark}
The twist hypothesis in this method of proof
can be bargained away considerably.
\end{remark}
\begin{remark} \label{rem:alternative}
A variant of the method that is useful in the study of
lower dimensional tori or for some degenerate situations,
is to take as a starting point of the procedure not just the $K$
but the $K$ and the $M$, which, respectively,
almost solve the equation and almost reduce the equation
to constant coefficients.
The iterative step, uses the $M$ to solve the
equation and then updates the $M$ so that it reduces
the new linearized equation to an even higher approximation.
By intertwining the improvement in $K$ and in $M$ it is
possible to achieve quadratic convergence.
One advantage of this improvement is that, if one studies this
for lower dimensional tori, both the $K$ and the $M$
can be computed perturbatively.
The approximate $K$ is a polynomial in the perturbation
parameter, nevertheless, the $M$ is a polynomial in the
square root. Hence, the iterative method based on both
approximations can capture the
singularity structure much better than the
approximation we have discussed here.
We refer to \cite{JorbaLZ00} for more details about the method
for lower dimensional tori.
\end{remark}
\begin{remark}
One feels that these methods of reducing the equation
to constant coefficients is a bit of overkill.
When one tries to invert an operator,
diagonalizing it is rather more than
what is needed.
Indeed, the great advances in KAM for PDE's
started when the emphasis went from
diagonalizing the linear operator as was done
usually in KAM theory to just using estimates
from the inverse. (See \cite{CraigW93},\cite{Bourgain95})
Even if we will not discuss it in these notes, when one
considers elliptic lower dimensional tori some of the
resonances that appear in some proofs are obstructions
to the diagonalization of the operator, not to the
invertibility. Therefore, they can be
eliminated from the proofs of the existence of the torus
if one relies on inverting the operator rather than just
diagonalizing it. (See \cite{Bourgain97}.)
Related to this issue we call attention to
the lectures of prof. Eliasson in this meeting.
He shows that even if it could happen that the
tori are not reducible, using his non-perturbative
results, they are arbitrarily close to reducible.
This is enough to continue the iterative procedure.
\end{remark}
\begin{remark}
One issue that still is quite puzzling to me is that
if one performs the Lindstedt series for lower
dimensional tori, one encounters only small denominators
related to the frequencies of the motion on the torus.
This is significantly less small denominators
than those appearing in the proofs
mentioned above in which one
needs to take into account denominators which
happen when harmonics of the intrinsic frequencies of
the torus are close to a normal frequency.
The proofs in which one also proves
reducibility of the lower dimensional
torus, require even more small denominators
conditions. In them,
one has to take into account the
cases when differences of two normal
frequencies become a combination of the frequencies of
motion in the torus.
In \cite{JorbaLZ00}, one can find a proof of the fact that the
Lindstedt series is asymptotic and defines an analytic function in
a large sector. (One has to exclude an exponentially thin wedge.)
Nevertheless, the convergence or not of these series has not
been settled.
Note that at the same time that one develops the series
for the torus, one develops also a series for the reducing matrix
which also does not present other small divisors than those of
the intrinsic frequencies. The convergence of this series
has not been settled either.
\end{remark}
\section{ Some remarks on computer assisted proofs}\label{compassisted}
The existence or non-existence of invariant tori in a system
appearing in a concrete application could have
enormous practical importance.
For example, there are many systems such as
accelerators or plasma devices that are modeled
rather well by Hamiltonian systems.
The existence of tori in these systems has
very drastic effects in their long term
behavior. For example, if the system is
a two dimensional map, the existence of
invariant circles, will imply that one region of
phase space will remain trapped for ever.
This is of great interest for plasma devices
whose goal is to confine a plasma or for
accelerators that try to keep a beam of particles in place.
Indeed, many of these devices are designed in such a
way that they maximize the {\sl abundance }
and robustness of invariant tori. One hopes that,
even if the Hamiltonian approximation is not
completely accurate, the KAM tori will survive somehow.
In celestial mechanics, one is interested also in
finding regions with invariant tori since they are
suitable for parking orbits.
The judicious numerical experimentation with
dynamical system has been a great source of
insight and inspiration, even if, of course,
much of the work is non-rigorous and, hence, does
not fit well with this tutorial.
We refer the reader to \cite{Henon83}, \cite{Simo98}
for some study of the issues involved in numerical
computations and to \cite{Meiss92} for
a point of view closer to the physical applications
Of course, in these applications, one also wants to get,
besides the existence, information about the shape of
the torus and more details about its properties.
What we want to discuss in this section is how some of
these non-rigorous calculations can be turned into theorems.
The basic observation is that
some of the KAM proofs
we have presented here have
the structure that they formulate a
functional equation and show that, given an approximate solution
which is not too bad from the analytic point of view, then
there is a true solution, which, moreover is not too
far from the approximate solutions.
These constructive methods do not require that the system
is close to integrable.
Note that these proofs do not care about how we have produced
the approximate solutions. The only thing that we need
to verify rigorously is that these approximate
solutions indeed solve the functional equation to
up to an small error
and that their analyticity properties are adequate.
Hence the problem of justifying that these computed
solution correspond to a true one
reduces to showing rigorously that these numerically
specified functions indeed solve the
desired functional equation with a good accuracy
and verifying rigorously their
analyticity properties.
Of course, given one polynomial with a few coefficients
one could imagine studying its
properties with respect to an easy
equation such as \eqref{lagrangianremainder}
with a pencil and a notepad.
(See \cite{Herman86} for an example of
these verifications with pencil and paper.)
Nevertheless, if the number of coefficients approaches
those needed for what is considered good accuracy in numerical
calculations (this is often a few hundred or a few thousand coefficients),
using a notepad becomes impossible.
One would like to use a computer.
The problem with using a computer is that, as they are used most
commonly computers do not deal with real numbers
and they do not perform on them the mathematical
arithmetic operations.
In their normal mode of working, computers
deal only
with a finite set of numbers, the {\sl representable} numbers
\begin{footnote}
In modern computers, there are almost
universally around $2^{64}$ representable numbers,
those which can be written in 8 bytes -- there are a few
delicate and complicated issues such as denormalized numbers.
Most computers also use for certain calculations
numbers with $80$ bits, which are, $2^80$ numbers.
There is a rather detailed standard by IEEE
\cite{IEEE85}
on how to perform arithmetic in numbers. It does
not only specify the precision to be used, but also
rounding and how to report troubles such as attempted
division by zero or overflow.
This standard is now almost universally implemented
in the chips and the languages
(rater inexplicably Java did not include it) and there are good tests of
compliance so that one can asses one's arithmetic.
See \cite{Kahan96}.
\end{footnote}
On these representable numbers, we perform arithmetic operations
which are approximations of the arithmetic operations
among real numbers.
These operations produce an approximation to the true answer
if at all possible
\begin{footnote}
The process of taking the true result and
producing a representable number is called {\sl rounding}.
Returning an representable number that is larger than the
true result is called rounding up, similarly
rounding down, rounding to nearest, rounding towards zero
etc.
The IEEE standard mentioned above specifies that the
user can control
the properties of the rounding and of the exceptions
by setting a control word.
\end{footnote}
or if it is impossible to give
a reasonable answer in terms of representable
numbers (e.g. if you ask to multiply by 10 the largest representative
number or to divide by
zero) they do not return an answer, but they {\sl
raise an exception} which typically does something drastic
such as causing the program to terminate abruptly, perhaps
copying the state of the memory to a file
({\sl dumping a core}) that can be examined to
trace the problem.
(A good discussion of the subtleties involved in the
implementation of floating point arithmetic is
\cite{Knuth97}.)
One problem with this approximate way of proceeding
is that approximate of
approximate may not be approximate enough.
Much less if one repeats the process of approximation
a large number of times. Of course, given that a computer
nowadays produces over one hundred million operations in
a second, we have to worry about the effect that
performing millions of approximations may lead
us away from a good approximation.
As every good numerical analyst knows, producing numbers is not too
difficult. Unless the computer catches fire, you will get
numbers. The real difficult issue is to produce numbers that can
be trusted. More difficult even is to device methods
that ensure the numbers produced can be trusted.
One should keep in mind that most of the technology
and research
happens at the borderline regions when the algorithms are about
to break. (If the problems we are
studying were safely solvable, we would fix the situation
going to a more challenging problem.)
The problem of reliability of arithmetic
calculations is significantly more pressing for the problems
involving small divisors. We have seen already that the
Lindstedt series manage to converge only through massive
amounts of cancellations. Cancellations are one of
the worst enemies of accuracy in floating point
calculations. Since computers
keep a fixed number of digits, adding numbers that
cancel almost exactly, will lead to a catastrophic lack
of precision (e.g. if we have 1.00001 and 1.00000
exact up to six digits, their difference will only have
one exact digit.)
Many of the problems with small divisors are such that the
numerics deteriorates in a complicated way until
the algorithms blow up or start behaving erratically.
\begin{exercise}
One of the standard programs to asses the characteristics
of a computer is
\begin{verbatim}
epsilon = 1.0;
oneplus = 1.0 + epsilon;
count = 0;
while (oneplus > 1.0){
epsilon /= 2.0;
oneplus = one + epsilon;
count++;
}
printf("%d", count);
\end{verbatim}
Run it in your computer.
Run also
\begin{verbatim}
epsilon = 1.0;
count = 0;
while ( 1.0 + epsilon > 1.0){
epsilon /= 2.0;
count++;
}
printf("%d", count);
\end{verbatim}
Chances are that the results will be quite different.
Explain why.
\end{exercise}
\begin{exercise}
The computer program Mathematica uses a
numerical scheme in which high precision numbers drop
precision if the last figures cannot be kept.
This leads to some
unexpected effects.
Run
\begin{verbatim}
a = N[Pi,40]
Do[ a = 2*a -a , {100}]
\end{verbatim}
and discuss the results.
\end{exercise}
One way of obtaining reliable results from a computer
without sacrificing too much performance is to
use {\sl interval arithmetic}
(See \cite{Moore79}, \cite{KaucherM84}.)
The idea is that a real variable is represented
by two representable numbers which are supposed to
mean an upper and a lower bound for the value of the
variable we are interested in.
Once one has bounds for the values of
a variable, one can operate on these bounds
in such a way that one always keeps obtaining bounds.
The only subtlety is that when adding upper bounds, one
has to round up, adding lower bounds, one has to round down,
etc. This can be done by reprogramming pieces of the
arithmetic, or, in systems that conform to the IEEE standard
by setting appropriately the control word.
This quickly leads to an arithmetic among intervals that
can produce bounds of arithmetic expressions given bounds
on the variables.
One can pass from bounds on arithmetic expressions
to bounds on sets in functional spaces. For example,
one can specify a set in
function space. For example, if
we specify a set of analytic functions by
\begin{equation} \label{representablesets}
U_{v_1, \ldots, v_n; \ep} = \{ f(z) \quad |\quad f(z) =
\sum_{i = 0}^N f_i z^i + f_e(z) , f_i \in v_i, || f_e||_1\le \ep \}
\end{equation}
where $v_i$ are intervals (i.e. pairs of representable
numbers) and $\ep$ is a representable number.
(There are, of course, many variants. One can
for example, take into account that some errors
are high order, use other norm for the error
or even several norms at the same time.)
It is reasonably easy to imagine how can one
define operations on sets of the type in \eqref{representablesets}
such that the numerical operations bound the real operations on
sets. With a bit more of imagination, one can do compositions,
integrals, and other operations. In particular, one can implement
the operations involved in the evaluation of the terms
in \eqref{linstedtlagrange2}.
If starting with the numerically produced non-rigorous guess
one can use the rigorous interval arithmetic to verify
the hypothesis of Theorem \ref{KAMlagrangian} -- or some other
theorem enjoying a similar structure -- then, one can guarantee
that there is a true solution near the computed one.
This strategy has been implemented in \cite{Rana87}, \cite{LlaveR90}.
Similar ideas have been implemented in
\cite{CellettiC95}.
Indeed, by now, starting with the inspiring proof of
\cite{Lanford82}
(it relied on the usual contraction mapping
theorem rather than in the hard implicit function theorems)
there has been a number of
significant theorems proved with similar techniques.
A survey of these developments is \cite{KochSW96}.
One of the main difficulties of the method is that it requires
to spend a great deal of time in
coding carefully the problems. One
can hope that some of the tasks could be automated
but there are difficulties. Even if automatic
translation of arithmetic expressions produces a valid answer,
arithmetic expressions that
are equivalent under the ordinary rules of arithmetic
are not equivalent under interval arithmetic.
For example in intervals
\begin{equation} \label{subdistributive}
(a + b) \times c \subset a \times c + b \times c
\end{equation}
and the inclusion can be strict.
A classic problem in interval arithmetic is to
find fast algorithms to compute accurately the image of
the unit disk under a polynomial.
\begin{exercise}
Give a proof of \eqref{subdistributive} and find
examples when it is strict.
\end{exercise}
I personally think that computer assisted proofs
and is a very interesting area in which
it is possible to find a meaningful collaboration between
Mathematicians (proving theorems of the right kind),
Computer Scientists (developing good software tools that relieve
the tedium of programming the variants required) and
applied scientists that have challenging real life problems.
\section{Acknowledgements}
The work of the author was supported in part
by NSF grants. I received substantial assistance in the
preparation of these notes from A. Haro, N. Petrov, J. Vano,
Parts of this
work are based on unpublished joint work
with other people that we intend to publish in fuller
versions.
Comments from H. Eliasson, T. Gramchev,
and many other participants in the SRI and by A. Jorba, M. Sevryuk, R.
Perez-Marco
removed many
mistakes and typos. Needless to say, they
are not to be blamed for those that escaped
their eye or for missing those that were introduced
in ulterior revisions which they did not see.
I also want
to express my appreciation of the great
amount of work put by
the AMS staff, specially W. Drady
and by A. Katok, Y. Pesin and, specially H.
Weiss to organize this Summer Institute.
It was a privilege and a humbling experience
to witness their dedication.
The enthusiasm of the participants in the
SRA was contagious.
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