\magnification=\magstep1
\def\real{{\bf R}} % "bbbr" is icm's style
\def\bT{{\bf T}}
\def\varep{{\varepsilon}}
% This is ICM.CMM the plain TeX macro package
% (CM version) from Springer-Verlag
% for the International Congress of Mathematicians 1990 in Kyoto
\font \tbfontt = cmbx10 scaled\magstep1
\font \tafontt = cmbx10 scaled\magstep2
\font \tbfontss = cmbx5 scaled\magstep1
\font \tafontss = cmbx5 scaled\magstep2
\font \sixbf = cmbx6
\font \tbfonts = cmbx7 scaled\magstep1
\font \tafonts = cmbx7 scaled\magstep2
\font \ninebf = cmbx9
\font \tasys = cmex10 scaled\magstep1
\font \tasyt = cmex10 scaled\magstep2
\font \sixi = cmmi6
\font \ninei = cmmi9
\font \tams = cmmib10
\font \tbmss = cmmib10 scaled 600
\font \tamss = cmmib10 scaled 700
\font \tbms = cmmib10 scaled 833
\font \tbmt = cmmib10 scaled\magstep1
\font \tamt = cmmib10 scaled\magstep2
\font \smallescriptscriptfont = cmr5
\font \smalletextfont = cmr5 at 10pt
\font \smallescriptfont = cmr5 at 7pt
\font \sixrm = cmr6
\font \ninerm = cmr9
\font \ninesl = cmsl9
\font \tensans = cmss10
\font \fivesans = cmss10 at 5pt
\font \sixsans = cmss10 at 6pt
\font \sevensans = cmss10 at 7pt
\font \ninesans = cmss10 at 9pt
\font \tbst = cmsy10 scaled\magstep1
\font \tast = cmsy10 scaled\magstep2
\font \tbsss = cmsy5 scaled\magstep1
\font \tasss = cmsy5 scaled\magstep2
\font \sixsy = cmsy6
\font \tbss = cmsy7 scaled\magstep1
\font \tass = cmsy7 scaled\magstep2
\font \ninesy = cmsy9
\font \markfont = cmti10 at 11pt
\font \nineit = cmti9
\font \ninett = cmtt9
%-----------------------------------------------------------------------
%\magnification=\magstep0
\hsize=12.2truecm
\vsize=19.4truecm
\hfuzz=2pt
\tolerance=500
\abovedisplayskip=3 mm plus6pt minus 4pt
\belowdisplayskip=3 mm plus6pt minus 4pt
\abovedisplayshortskip=0mm plus6pt minus 2pt
\belowdisplayshortskip=2 mm plus4pt minus 4pt
\predisplaypenalty=0
\clubpenalty=10000
\widowpenalty=10000
\frenchspacing
\newdimen\oldparindent\oldparindent=1.5em
\parindent=1.5em
%-----------------------------------------------------------------------
\def\getsto{\mathrel{\mathchoice {\vcenter{\offinterlineskip
\halign{\hfil
$\displaystyle##$\hfil\cr\gets\cr\to\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr\gets
\cr\to\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr\gets
\cr\to\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
\gets\cr\to\cr}}}}}
\def\lid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle##$\hfil\cr<\cr\noalign{\vskip1.2pt}=\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr
\noalign{\vskip1.2pt}=\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr
\noalign{\vskip1pt}=\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
<\cr
\noalign{\vskip0.9pt}=\cr}}}}}
\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2pt}=\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr
\noalign{\vskip1.2pt}=\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr
\noalign{\vskip1pt}=\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
>\cr
\noalign{\vskip0.9pt}=\cr}}}}}
\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip
\halign{\hfil
$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-1pt}<\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr
>\cr\noalign{\vskip-1pt}<\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr
>\cr\noalign{\vskip-0.8pt}<\cr}}}
{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
>\cr\noalign{\vskip-0.3pt}<\cr}}}}}
\def\bbbr{{\rm I\!R}} %reelle Zahlen
\def\bbbm{{\rm I\!M}}
\def\bbbn{{\rm I\!N}} %natuerliche Zahlen
\def\bbbf{{\rm I\!F}}
\def\bbbh{{\rm I\!H}}
\def\bbbk{{\rm I\!K}}
\def\bbbp{{\rm I\!P}}
\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}}
\def\bbbe{{\mathchoice {\setbox0=\hbox{\smalletextfont e}\hbox{\raise
0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.3pt
height0.7\ht0\hss}\box0}}
{\setbox0=\hbox{\smalletextfont e}\hbox{\raise 0.1\ht0\hbox
to0pt{\kern0.4\wd0\vrule width0.3pt height0.7\ht0\hss}\box0}}
{\setbox0=\hbox{\smallescriptfont e}\hbox{\raise 0.1\ht0\hbox
to0pt{\kern0.5\wd0\vrule width0.2pt height0.7\ht0\hss}\box0}}
{\setbox0=\hbox{\smallescriptscriptfont e}\hbox{\raise
0.1\ht0\hbox to0pt{\kern0.4\wd0\vrule width0.2pt
height0.7\ht0\hss}\box0}}}}
\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm Q$}\hbox{\raise
0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise
0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise
0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise
0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}}
\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox
to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox
to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox
to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}}
\def\bbbs{{\mathchoice
{\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox
to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox
to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox
to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox
to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}}
\def\bbbz{{\mathchoice {\hbox{$\sans\textstyle Z\kern-0.4em Z$}}
{\hbox{$\sans\textstyle Z\kern-0.4em Z$}}
{\hbox{$\sans\scriptstyle Z\kern-0.3em Z$}}
{\hbox{$\sans\scriptscriptstyle Z\kern-0.2em Z$}}}}
%-----------------------------------------------------------------------
% petit-fonts
\skewchar\ninei='177 \skewchar\sixi='177
\skewchar\ninesy='60 \skewchar\sixsy='60
\hyphenchar\ninett=-1
\def\newline{\hfil\break}%
%-----------------------------------------------------------------------
\catcode`@=11
\def\folio{\ifnum\pageno<\z@
\uppercase\expandafter{\romannumeral-\pageno}%
\else\number\pageno \fi}
\catcode`@=12 % at signs are no longer letters
%-------------------------------------------------------
% Definition der versal griechischen Buchstaben
%=======================================================================
\mathchardef\Gamma="0100
\mathchardef\Delta="0101
\mathchardef\Theta="0102
\mathchardef\Lambda="0103
\mathchardef\Xi="0104
\mathchardef\Pi="0105
\mathchardef\Sigma="0106
\mathchardef\Upsilon="0107
\mathchardef\Phi="0108
\mathchardef\Psi="0109
\mathchardef\Omega="010A
%-----------------------------------------------------------------------
\def\squareforqed{\hbox{\rlap{$\sqcap$}$\sqcup$}}
\def\qed{\ifmmode\squareforqed\else{\unskip\nobreak\hfil
\penalty50\hskip1em\null\nobreak\hfil\squareforqed
\parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi}
%-----------------------------------------------------------------------
\newfam\sansfam
\textfont\sansfam=\tensans\scriptfont\sansfam=\sevensans
\scriptscriptfont\sansfam=\fivesans
\def\sans{\fam\sansfam\tensans}
%-----------------------------------------------------------------------
\def\stackfigbox{\if
Y\FIG\global\setbox\figbox=\vbox{\unvbox\figbox\box1}%
\else\global\setbox\figbox=\vbox{\box1}\global\let\FIG=Y\fi}
%
\def\placefigure{\dimen0=\ht1\advance\dimen0by\dp1
\advance\dimen0by5\baselineskip
\advance\dimen0by0.4true cm
\ifdim\dimen0>\vsize\pageinsert\box1\vfill\endinsert
\else%keine seitenhohe Abbildung
\if Y\FIG\stackfigbox\else
\dimen0=\pagetotal\ifdim\dimen0<\pagegoal%akt. Seite ist noch nicht voll
\advance\dimen0by\ht1\advance\dimen0by\dp1\advance\dimen0by1.7true cm
\ifdim\dimen0>\pagegoal\stackfigbox
\else\box1\vskip7true mm\fi
\else\box1\vskip7true mm\fi\fi\fi\let\firstleg=Y}
%
% Abbildungen
\def\begfig#1cm#2\endfig{\par
\setbox1=\vbox{\dimen0=#1true cm\advance\dimen0
by1true cm\kern\dimen0\vskip-.8333\baselineskip#2}\placefigure}
%
\def\begdoublefig#1cm #2 #3 \enddoublefig{\begfig#1cm%
\line{\vtop{\hsize=0.46\hsize#2}\hfill
\vtop{\hsize=0.46\hsize#3}}\endfig}
%-------------------------------------------------------------------
\let\firstleg=Y
% Abbildungslegenden
% Falls Text kleiner als eine volle Zeile, zentriert.
\def\figure#1#2{\if Y\firstleg\vskip1true cm\else\vskip1.7true mm\fi
\let\firstleg=N\setbox0=\vbox{\noindent\petit{\bf
Fig.\ts#1\unskip.\ }\ignorespaces #2\smallskip
\count255=0\global\advance\count255by\prevgraf}%
\ifnum\count255>1\box0\else
\centerline{\petit{\bf Fig.\ts#1\unskip.\
}\ignorespaces#2}\smallskip\fi}
%-----------------------------------------------------------------
% Tabellenkoepfe
\def\tabcap#1#2{\smallskip\vbox{\noindent\petit{\bf Table\ts#1\unskip.\
}\ignorespaces #2\medskip}}
%-------------------------------------------------------------------
\def\begtab#1cm#2\endtab{\par
\ifvoid\topins\midinsert\medskip\vbox{#2\kern#1true cm}\endinsert
\else\topinsert\vbox{#2\kern#1true cm}\endinsert\fi}
%-------------------------------------------------------------------
\def\begpet{\vskip6pt\bgroup\petit}
\def\endpet{\vskip6pt\egroup}
%-------------------------------------------------------------------
% Referenzen
\newdimen\refindent
\newlinechar=`\|
\def\begref#1#2{\titlea{}{#1}%
\bgroup\petit
\setbox0=\hbox{#2\enspace}\refindent=\wd0\relax
\if!#2!\else
\ifdim\refindent>0.5em\else
\message{|Something may be wrong with your references;}%
\message{probably you missed the second argument of \string\begref.}%
\fi\fi}
\def\ref{\goodbreak
\hangindent\oldparindent\hangafter=1
\noindent\ignorespaces}
\def\refno#1{\goodbreak
\setbox0=\hbox{#1\enspace}\ifdim\refindent<\wd0\relax
\message{|Your reference `#1' is wider than you pretended in using
\string\begref.}\fi
\hangindent\refindent\hangafter=1
\noindent\kern\refindent\llap{#1\enspace}\ignorespaces}
\def\refmark#1{\goodbreak
\setbox0=\hbox{#1\enspace}\ifdim\refindent<\wd0\relax
\message{|Your reference `#1' is wider than you pretended in using
\string\begref.}\fi
\hangindent\refindent\hangafter=1
\noindent\hbox to\refindent{#1\hss}\ignorespaces}
\def\endref{\goodbreak\endpet}% Ende der Referenzen
%-------------------------------------------------------------------
\def\vec#1{{\textfont1=\tenbf\scriptfont1=\sevenbf
\textfont0=\tenbf\scriptfont0=\sevenbf
\mathchoice{\hbox{$\displaystyle#1$}}{\hbox{$\textstyle#1$}}
{\hbox{$\scriptstyle#1$}}{\hbox{$\scriptscriptstyle#1$}}}}
%---------------------------------------------------------------------
\def\petit{\def\rm{\fam0\ninerm}%
\textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
\textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei
\textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
\def\it{\fam\itfam\nineit}%
\textfont\itfam=\nineit
\def\sl{\fam\slfam\ninesl}%
\textfont\slfam=\ninesl
\def\bf{\fam\bffam\ninebf}%
\textfont\bffam=\ninebf \scriptfont\bffam=\sixbf
\scriptscriptfont\bffam=\fivebf
\def\sans{\fam\sansfam\ninesans}%
\textfont\sansfam=\ninesans \scriptfont\sansfam=\sixsans
\scriptscriptfont\sansfam=\fivesans
\def\tt{\fam\ttfam\ninett}%
\textfont\ttfam=\ninett
\normalbaselineskip=11pt
\setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}%
\normalbaselines\rm
\def\vec##1{{\textfont1=\tbms\scriptfont1=\tbmss
\textfont0=\ninebf\scriptfont0=\sixbf
\mathchoice{\hbox{$\displaystyle##1$}}{\hbox{$\textstyle##1$}}
{\hbox{$\scriptstyle##1$}}{\hbox{$\scriptscriptstyle##1$}}}}}
%-------------------------------------------------------------------
\nopagenumbers
%
% Der Schalter \header gibt an, ob ein "running head" gedruckt werden
% soll; wenn er nicht auf "N" steht kommt ein solcher.
\let\header=Y
\let\FIG=N
\newbox\figbox
\output={\if N\header\headline={\hfil}\fi\plainoutput
\global\let\header=Y\if Y\FIG\topinsert\unvbox\figbox\endinsert
\global\let\FIG=N\fi}
%------------------------------------------------------
\let\lasttitle=N
%---------------------------------------------------------------
\catcode`\@=\active
\def\author#1{\bgroup
\baselineskip=13.2pt
\lineskip=0pt
\pretolerance=10000
\markfont
\ignorespaces#1\bigskip\egroup
{\def@##1{}%
\setbox0=\hbox{\petit\kern2.5true cc\ignorespaces#1\unskip}%
\ifdim\wd0>\hsize
\message{The names of the authors exceed the headline, please use a }%
\message{short form with AUTHORRUNNING}\gdef\leftheadline{%
\hbox to2.5true cc{\folio\hfil}\hfil AUTHORS suppressed due to excessive
length}%
\else
\xdef\leftheadline{\hbox to2.5true
cc{\noexpand\folio\hfil}\hfill\ignorespaces#1\unskip}%
\fi
}\let\INS=E}
\def\address#1{\bgroup\petit
\ignorespaces#1\bigskip\egroup
\catcode`\@=12
\vskip2cm\noindent\ignorespaces}
%---------------------------------------------------------------------
\let\INS=N%
% Aktionen, die bei Antreffen des @-Zeichens zu machen sind;
% drei Faelle a) @ bei AUTHOR, b) 1.@ bei ADDRESS, c) alle weiteren @'s
\def@#1{\if N\INS\unskip$\,^{#1}$\else\global\footcount=#1\relax
\if E\INS\hangindent0.5\parindent\noindent\hbox
to0.5\parindent{$^{#1}$\hfil}\let\INS=Y\ignorespaces
\else\par\hangindent0.5\parindent\noindent\hbox
to0.5\parindent{$^{#1}$\hfil}\ignorespaces\fi\fi}%
\catcode`\@=12
%-------------------------------------------------------------------
% "running head"
\headline={\petit\def\newline{ }\def\fonote#1{}\ifodd\pageno
\rightheadline\else\leftheadline\fi}
\def\rightheadline{Missing CONTRIBUTION
title\hfil\hbox to2.5true cc{\hfil\folio}}
\def\leftheadline{\hbox to2.5true cc{\folio\hfil}\hfil Missing name(s)
of the author(s)}
\nopagenumbers
%
\let\header=Y
%------------------------------------------------------
\def\contributionrunning#1{\message{Running head on right hand sides
(CONTRIBUTION)
has been changed}\gdef\rightheadline{\ignorespaces#1\unskip\hfil
\hbox to2.5true cc{\hfil\folio}}}
\def\authorrunning#1{\message{Running head on left hand sides (AUTHOR)
has been changed}\gdef\leftheadline{\hbox to2.5true cc{\folio
\hfil}\hfil\ignorespaces#1\unskip}}
%------------------------------------------------------
\let\lasttitle=N
\def\contribution#1{\vfill\eject
\let\header=N\bgroup
\textfont0=\tafontt \scriptfont0=\tafonts \scriptscriptfont0=\tafontss
\textfont1=\tamt \scriptfont1=\tams \scriptscriptfont1=\tams
\textfont2=\tast \scriptfont2=\tass \scriptscriptfont2=\tasss
\par\baselineskip=16pt
\lineskip=16pt
\tafontt
\raggedright
\pretolerance=10000
\noindent
\ignorespaces#1
\vskip17pt\egroup
\nobreak
\parindent=0pt
\everypar={\global\parindent=1.5em
\global\let\lasttitle=N\global\everypar={}}%
\global\let\lasttitle=A%
\setbox0=\hbox{\petit\def\newline{ }\def\fonote##1{}\kern2.5true
cc\ignorespaces#1}\ifdim\wd0>\hsize
\message{Your CONTRIBUTIONtitle exceeds the headline,
please use a short form
with CONTRIBUTIONRUNNING}\gdef\rightheadline{CONTRIBUTION title
suppressed due to excessive length\hfil\hbox to2.5true cc{\hfil\folio}}%
\else
\gdef\rightheadline{\ignorespaces#1\unskip\hfil\hbox to2.5true
cc{\hfil\folio}}\fi
\catcode`\@=\active
\ignorespaces}
%------------------------------------------------------
% Beginn Ueberschrift 1. Ordnung
\def\titlea#1#2{\if N\lasttitle\else\vskip-28pt
\fi
\vskip18pt plus 4pt minus4pt
\bgroup
\textfont0=\tbfontt \scriptfont0=\tbfonts \scriptscriptfont0=\tbfontss
\textfont1=\tbmt \scriptfont1=\tbms \scriptscriptfont1=\tbmss
\textfont2=\tbst \scriptfont2=\tbss \scriptscriptfont2=\tbsss
\textfont3=\tasys \scriptfont3=\tenex \scriptscriptfont3=\tenex
\baselineskip=16pt
\lineskip=0pt
\pretolerance=10000
\noindent
\tbfontt
\rightskip 0pt plus 6em
\setbox0=\vbox{\vskip23pt\def\fonote##1{}%
\noindent
\if!#1!\ignorespaces#2
\else\ignorespaces#1\unskip\enspace\ignorespaces#2\fi
\vskip18pt}%
\dimen0=\pagetotal\advance\dimen0 by-\pageshrink
\ifdim\dimen0<\pagegoal
\dimen0=\ht0\advance\dimen0 by\dp0\advance\dimen0 by
3\normalbaselineskip
\advance\dimen0 by\pagetotal
\ifdim\dimen0>\pagegoal\eject\fi\fi
\noindent
\if!#1!\ignorespaces#2
\else\ignorespaces#1\unskip\enspace\ignorespaces#2\fi
\vskip12pt plus4pt minus4pt\egroup
\nobreak
\parindent=0pt
\everypar={\global\parindent=\oldparindent
\global\let\lasttitle=N\global\everypar={}}%
\global\let\lasttitle=A%
\ignorespaces}
%------------------------------------------------------
% Beginn Ueberschrift 2. Ordnung
\def\titleb#1#2{\if N\lasttitle\else\vskip-22pt
\fi
\vskip18pt plus 4pt minus4pt
\bgroup
\textfont0=\tenbf \scriptfont0=\sevenbf \scriptscriptfont0=\fivebf
\textfont1=\tams \scriptfont1=\tamss \scriptscriptfont1=\tbmss
\lineskip=0pt
\pretolerance=10000
\noindent
\bf
\rightskip 0pt plus 6em
\setbox0=\vbox{\vskip23pt\def\fonote##1{}%
\noindent
\if!#1!\ignorespaces#2
\else\ignorespaces#1\unskip\enspace\ignorespaces#2\fi
\vskip10pt}%
\dimen0=\pagetotal\advance\dimen0 by-\pageshrink
\ifdim\dimen0<\pagegoal
\dimen0=\ht0\advance\dimen0 by\dp0\advance\dimen0 by
3\normalbaselineskip
\advance\dimen0 by\pagetotal
\ifdim\dimen0>\pagegoal\eject\fi\fi
\noindent
\if!#1!\ignorespaces#2
\else\ignorespaces#1\unskip\enspace\ignorespaces#2\fi
\vskip8pt plus4pt minus4pt\egroup
\nobreak
\parindent=0pt
\everypar={\global\parindent=\oldparindent
\global\let\lasttitle=N\global\everypar={}}%
\global\let\lasttitle=B%
\ignorespaces}
%------------------------------------------------------
% Beginn Ueberschrift 3. Ordnung
\def\titlec#1{\if N\lasttitle\else\vskip-\baselineskip
\fi
\vskip18pt plus 4pt minus4pt
\bgroup
\textfont0=\tenbf \scriptfont0=\sevenbf \scriptscriptfont0=\fivebf
\textfont1=\tams \scriptfont1=\tamss \scriptscriptfont1=\tbmss
\bf
\noindent
\ignorespaces#1\unskip\ \egroup
\ignorespaces}
%-------------------------------------------------------------------
% Beginn Ueberschrift 4. Ordnung
\def\titled#1{\if N\lasttitle\else\vskip-\baselineskip
\fi
\vskip12pt plus 4pt minus 4pt
\bgroup
\it
\noindent
\ignorespaces#1\unskip\ \egroup
\ignorespaces}
%-------------------------------------------------------------------
\let\ts=\thinspace
\def\footnoterule{\kern-3pt\hrule width 2true cm\kern2.6pt}
% Fussnoten-macros
\newcount\footcount \footcount=0
\def\advftncnt{\advance\footcount by1\global\footcount=\footcount}
% Automatisch numerierte Fussnote, Fussnotentex in petit
\def\fonote#1{\advftncnt$^{\the\footcount}$\begingroup\petit
\parfillskip=0pt plus 1fil
\def\textindent##1{\hangindent0.5\oldparindent\noindent\hbox
to0.5\oldparindent{##1\hss}\ignorespaces}%
\vfootnote{$^{\the\footcount}$}{#1\vskip-9.69pt}\endgroup}
%-------------------------------------------------------------------
\def\item#1{\par\noindent
\hangindent6.5 mm\hangafter=0
\llap{#1\enspace}\ignorespaces}
%-------------------------------------------------------------------
\def\itemitem#1{\par\noindent
\hangindent11.5 mm\hangafter=0
\llap{#1\enspace}\ignorespaces}
%-------------------------------------------------------------------
\def\newenvironment#1#2#3#4{\long\def#1##1##2{\removelastskip
\vskip\baselineskip\noindent{#3#2\if!##1!.\else\unskip\ \ignorespaces
##1\unskip\fi\ }{#4\ignorespaces##2}\vskip\baselineskip}}
% Lemma, Proposition, Theorem, Corollary
\newenvironment\lemma{Lemma}{\bf}{\it}
\newenvironment\proposition{Proposition}{\bf}{\it}
\newenvironment\theorem{Theorem}{\bf}{\it}
\newenvironment\corollary{Corollary}{\bf}{\it}
%---------------------------------------------------------------------
% Example, Exercise, Problem, Solution, Definition
\newenvironment\example{Example}{\it}{\rm}
\newenvironment\exercise{Exercise}{\bf}{\rm}
\newenvironment\problem{Problem}{\bf}{\rm}
\newenvironment\solution{Solution}{\bf}{\rm}
\newenvironment\definition{Definition}{\bf}{\rm}
%---------------------------------------------------------------------
%Note, Question
\newenvironment\note{Note}{\it}{\rm}
\newenvironment\question{Question}{\it}{\rm}
%---------------------------------------------------------------------
%Proof, Remark
\long\def\remark#1{\removelastskip\vskip\baselineskip\noindent{\it
Remark.\ }\ignorespaces}
\long\def\proof#1{\removelastskip\vskip\baselineskip\noindent{\it
Proof\if!#1!\else\ \ignorespaces#1\fi.\ }\ignorespaces}
%------------------------------------------------------------------
\def\typeset{\petit\noindent This article was processed by the author
using the \TeX\ macro package from Springer-Verlag.\par}
\outer\def\byebye{\bigskip\bigskip\typeset
\footcount=1\ifx\speciali\undefined\else
\loop\smallskip\noindent special character No\number\footcount:
\csname special\romannumeral\footcount\endcsname
\advance\footcount by 1\global\footcount=\footcount
\ifnum\footcount<11\repeat\fi
\vfill\supereject\end}
\overfullrule=0pt
\hsize 6.5 true in
\vsize 9 true in
\contribution{Recent Progress in Classical Mechanics}
\author{R. de la Llave}
\address{Department of Mathematics,
The University of Texas at Austin,
Austin, Texas 78712 USA}
The goal of this lecture is to review several developments in classical
mechanics that have taken place in the last years, that will fit in the
time of the talk and that I have become aware of. Unfortunately, the
latter is a constraint more severe than what I would like and I
apologize to the authors and the audience for many things that have been
left out.
In particular, I have left out topics such as ``twist mappings'',
or
``geometric phases''
and ``quantum chaos'' that are generating a great deal ofactivity in the
literature.
\titlea{}{Geometric theory of integrable systems}
One of the central problems of mechanics has been to integrate Hamilton's
equations of motion, or at least decide if such an integration is impossible.
The most geometrically natural notion of integrability is that the
system should have as many conserved quantities with vanishing Poisson
brackets as degrees of freedom.
It has long been known that the fact that a system is integrable
severely restricts
the topology of the phase space and of the energy surface. For
example, if the system admits action-angle variables --- which is strictly
stronger than being integrable in the above sense --- the phase space should
be $\real^n \times \bT^n$ so that the topology is determined.
The first foothold in the geometric theory of integrable systems is the
Liouville/Arnol'd theorem that says that the system can be decomposed
in pieces $\real^{n_i} \times \bT^{m_i}$, but the $n_i,m_i$ can change.
For example in the Kepler system we have bounded trajectories along
ellipses and unbounded ones along hyperbolas. These two sets get
``glued'' in the intermediate set of parabolic trajectories.
An important realization was that
the gluing of the different pieces has to be done in very precise ways
so as to preserve the very rigid structure imposed by integrability.
Hence, the phase space of an integrable system can be considered as
pieces of $\real^{n_i} \times \bT^{m_i}$ glued in very precise ways.
This turns out to impose severe restrictions on the phase spaces and energy
surfaces of integrable systems. (Notice that having standard pieces that
get glued in well defined ways according to the singular submanifolds
of a vector field is somewhat reminiscent of Morse Theory.)
This beautiful theory, which one may start to learn in the paper
[Fo] written by the indefatigable leader of the method, has many
spin-offs.
Since one of the best known methods to generate
integrable systems has been to
create systems in manifolds with lots of symmetries --- e.g., groups
or quotients of groups --- it is possible to obtain many results about
topological consequences of algebraic structures. The machinery of cutting
and pasting manifolds while preserving a rich geometric structure can be
used to generate examples and counterexamples in low dimensional topology.
One of the methods of choice to attempt the classification of low dimensional
manifolds has been to show that on them one can define canonical
geometric structures that produce invariants, and, one hopes, provide
with methods to show they are equivalent.
Once one has an integrable system on a manifold, it is very easy to
generate other types of structures. Hence, these methods have tested
the boundaries of what geometric structures determine.
\titlea{}{Analytically integrable systems}
On the other side of the spectrum one can try to decide which
systems can be integrated using only algebraic or rational transformations.
For example, if one takes particles on a line which are very repulsive,
the resulting system will be integrable [Hub] [Gu]--- essentially, the
integration is achieved by the scattering operator --- nevertheless, the
Calogero potential is special because it can be integrated by algebraic
methods.
There are several criteria to show that a system cannot be integrated
by algebraic methods. The most time honored is the one used by
S.~Kowaleska to narrow down the search of algebraically integrable
cases for the rigid body to a few particular ones. (Basically, we
observe that an integrated system has no singularities and that an
algebraic integration can only introduce algebraic singularities. The
possible singularities of the system under study can be ascertained
by solving the differential equation.)
A more recent method for complex integrability has been introduced by
Ziglin, based on the observation that if a system is integrable,
the periodic orbits can be deformed, along complex paths. This imposes
contraints on the algebraic properties of the variational equations.
To my knowledge, there are no definitive results about converses of these
negative criteria. One would like to know if an algebraic system
satisfying Kowaleska criterion, and maybe some other global condition,
is algebraically integrable.
A much more severe obstruction to the possibility of integrability
is the existence of homoclinic points. The verification of this hypothesis
usually is done by perturbation theory. We mention that in several cases,
this is quite difficult since the perturbation expansions vanishes to all
orders.
A very easy to read review of the developments in the physical literature
is [RGB]. See also the corresponding chapters in [AANS], [Koz], [Mor].
One could wonder why one should worry about the difference between
algebraic integrability versus, say $C^{50}$.
Let me mention two reasons. One is that there are several structures
that only manifest themselves when we consider complex extensions.
(We will discuss one of those when we discuss perturbations beyond
all orders.) Another is that there are many natural systems that
are algebraic, for example, one step of algorithms acting on matrices
can be considered as an algebraic dynamical system. It has been known
for a long time that there were important similarities between the
Jacobi algorithm to diagonalize matrices and the scattering of
particles on a line. This has been used to analyze in quite detailed
fashion the Jacobi algorithm and in turn this has lead to implementations
that soon will improve in many respects the now standard
algorithms [DLT].
>From the mathematical point of view, the theory of algebraically
integrable systems has important connections with algebraic geometry.
For example, Hilbert's $21^{\rm st}$ problem is related to issues raised
by Ziglin's method.
\titlea{}{Asymptotics beyond all orders}
There are several interesting quantities in classical mechanics that,
do not vanish even if a perturbation expansion yields identically zero.
It has been known for a long time these perturbation expansions
(physicists called them ``divergent'' even if they are the sum of zero
term) into upper bounds. In a typical situation, by truncating the
series to order $N$ we obtain $|Q| \le \varep^N N^N$. Then, if
we take $N= (\varep e)^{-1}$ we obtain $|Q| \le e^{-1/\varep e}$.
Such upper bounds appear very frequently. See for example
[Nek] for transport, [Ne2], [FS] for splitting of
separatrices.
Getting lower bounds seems much harder. Nevertheless, there is a trick that
has yielded results in several cases. If we consider complex extensions
of radii $O(1/\varep)$ for the objects under consideration,
perturbation theory will produce results
$Q(x) = A(x)\varep + O(\varep^2)$. Then, using Cauchy estimates, it is
possible to prove lower bounds of the right order of magnitude.
Needless to say, this broad sketch does not do justice to the difficulty
of concrete situations. Some recent examples are the treatment of the
Landau-Zener formula for the corrections to the adiabatic limit [Ha], [JaSe]
or the crossing of separatrices in two dimensional twist mappings, [An],
[ACKR] a problem that has plagued the literature for several years.
Let us point out that the upper bounds in [Nek] do not have a
corresponding set of lower bounds. This is the famous problem of Arnol'd
diffusion and it involves, besides analytical lower bounds, a good
understanding of several geometric structures.
It is known that this phenomenon occurs in examples [Ar1]
and there are now more systematic constructions [Do] that show it is
generic. Nevertheless, this is a far cry from being able to decide
whether a concrete system presents it or not or calculate its magnitude.
Let us point out that in this problem of Nekhorosev bounds/Arnol'd diffusion
there has been significant recent progress. For upper bounds, the
paper [Nek] was significantly clarified in [BGG] and a radically different
proof has appeared [Lo]. For Arnol'd diffusion, there has been
progress in the computation of one of the ingredients, the whiskered
tori. There are two different perturbative calculations of whiskered tori
that yield disjoint sets of tori [LW], [Tr].
\titlea{}{Non-collision singularities in the $N$-body problem}
The problem of $N$-bodies moving classically under their mutual
gravitational attraction is perhaps the oldest in mathematical physics.
The first question to ask is whether solutions are defined for all time.
A first glance to the differential equations reveals that they become
singular if two bodies collide. Nevertheless, a more detailed analysis
reveal that the singularity when only two bodies collide
is only apparent. If we modify the conditions
so that the bodies miss by a small amount, we get a well defined limit
as this modification goes to zero (a back of the envelope argument: the
conservation of momentum and energy determines what happens). Unfortunately,
for triple collisions, this argument does not work and indeed, it is possible
to show that if the bodies miss by small amounts many things can happen and
one does not get a well defined limit (exercise: show that the same
happens with billiard balls). Very detailed information is
nevertheless available (see [SM], ch.I for the the classical work of
Sundmann, [De]) for the three body collision and glimpses for more bodies.
One natural question to ask is how pervasive these singularities are and
whether there are any others. Important results were obtained
at the beginning of the century and completed in the
early 70's (See [SM] ) that
showed that if a solution cannot be continued, either
there is a triple collision or the moment of inertia of
the whole system has to become unbounded.
In [MMcG] it was shown that, indeed there are singularities
different from collisions.
Their example consists of 4 particles in a line. Two of them are
oscillating and a third one is far apart. A ``messenger particle''
collides with the oscillating pair and, since this is almost a triple
collision, leaves at an enormous speed obtained from the potential
energy of the pair. It catches the other particle, gives momentum
to it and bounces back, just in time to catch the other two in an almost
triple collision, in precisely the conditions that will make it
bounce back
even more violently and so on. The net result
is that the fourth particle escapes to infinity in finite time.
Unfortunately, it was quite difficult to generalize their method to
higher dimension so as to avoid collisions completely. (This was
included in the list of problems in mathematical physics by B.~Simon.)
Recently, however, there have been two remarkable papers [Xi], [Ge] in
which examples with collisionless singularities
are produced. Both are based on having
a messenger particle going back and forth between others managing
to
always
arrive
near a triple collision (but not colliding).
The configuration in [Xi] has two pairs rotating on parallel planes
directly above each other
and the messenger particle moving perpendicularly. The configuration of [Ge]
consists of pairs orbiting around --- roughly the vertices of
an equilateral polygon and the several
messenger particles going around them.
These two papers entail quite involved estimates and are somewhat
difficult to read (I have not checked them in detail myself) but it is clear that
they are important.
Let me point out that it is not known whether such singularities occupy
a set of positive measure.
The most obvious quantum version of the problem of showing that the dynamics is
well defined for all times for almost all trajectories is to show that the
quantum Hamiltonian is self-adjoint, which is much easier than the classical
one. (See, nevertheless, [RS] for some
more detailed physical discussion of the
analogy) It is amusing to note that the effect of a messenger particle
oscillating wildly between two channels is, however, the enemy to beat
in the proofs of quantum asymptotic completeness.
\titlea{}{Examples of classical ergodic systems}
The main argument to prove that a system is ergodic originated in the work of
[Hed], [Ho] (the latter published in Leipzig!) which
established ergodicity of
geodesic flows.
Basically the main ingredients are a geometric study of the trajectories,
which establishes that trajectories that converge in the future or the past
form a manifold and that by moving alternatively along these
stable/unstable manifolds we can go from every point to every point.
Secondly, an abstract theorem due to Birkhoff that shows that for almost
all trajectories, the statistical behaviour in the future is the same as
that in the past. (Unfortunately, due to the ``almost all'' rather than all
in Birkhoff theorem we need an extra technical condition on the
foliations.)
The argument was generalized and streamlined in [A].
Further generalizations were due to [Si] who introduced allowing
singularities and [Pe] who allowed for non-uniformity of the
approaches in the future and the past but constructed the
stable and unstable manifolds and concluded that,
if they are sufficiently long, the system is indeed ergodic,
if they are not, there are counterexamples [Pe2], [W].
Nevertheless, one can say that, in spite of its beauty, the theory
was too abstract and the
the only concrete examples with physical appeal
were the dispersing billiards [Si], joined later by the
celebrated stadium [Bu], a non-dispersing ergodic billiard.
Recently, the situation has changed,
manageable conditions to verify that concrete systems satisfy the abstract
hypothesis of [Pe] were introduced in [W]. This unified
the examples of dispersing and non-dispersing billiards
and examples of ergodic
geodesic flows and flows on scattering potentials
were constructed [Don], [DoL].
More or less at the same time, there appeared another elaboration
of the basic strategy that can produce ergodicity even in the case
that the manifolds are short [SCh] or that there are singularities
The basic idea is perhaps the concept of local ergodicity.
The methods of this paper have been extended in [KSS].
Let us mention that all the above verifications
of ergodicity, almost automatically yield ergodic properties such as
K-property, positive entropy, Bernouilli.
Maybe I should mention that presumably proving ergodicity is not quite
the only problem one wants to tackle
for physical applications. For Hamiltonian systems, it is very
easy to create little islands that, even if they will be
negligible for practical applications,
would destroy ergodicity. A very interesting problem that I
learned from T.~Spencer
is to prove that the well known system
$T_\varep (A,\theta) = (A+\varep \sin\theta, \theta +A+\varep \sin\theta)$
has an
ergodic component of positive measure for some large $\varep$'s.
Let me mention that if one takes in place of sin a piecewise
linear version
$$f(x) = \cases{
x&if $x\in [0,\pi/2]$\cr
\pi-x&if $x\in [\pi/2,\, 3\pi/2]$\cr
x-2\pi&if $x\in [3\pi/2,\, 2\pi]$\cr}$$
It is reasonably easy to verify hypothesis
that imply that the piecewise linear
version of $T$ is ergodic for $\varep >100$. Can one make a
prove the same result for some values of $\varep$
when we round the corners of the picewise liner version of
$T$?.
\titlea{}{Rigidity of dynamical systems}
An object is called rigid if, whenever there is another object
equivalent to it in a certain sense, it is also
equivalent in another stronger sense.
For example, a triangle is rigid because any polygon equivalent to it
in the sense of having sides of the same length is equivalent in
the much stronger sense of being isometric.
A somewhat weaker version of rigidity is rigidity under deformations.
We say that a
system is rigid under deformations
if all the deformations
that preserve some structure are trivial.
Typically when one starts working with equivalences
one tries to attach invariants and it is usual to also call a rigidity
theorem a result that shows that some set of invariants determines
the object up to trivial changes.
One of the reasons why rigidity theorems are amusing to study is
that they cross category lines. We investigate
measure theoretic consequences of a
Riemannian hypothesis and so on.
One set of objects for which many rigidity theorems are known
is negatively curved Riemannian manifolds.
A very interesting line of development was started in the papers
[GK1][GK2] which showed that,
for negatively curved manifolds
(either two dimensional or satisfying pinching conditions),
isospectral deformations of the metric are isometries
(this is a version of the famous
``can you hear the shape of a drum?''
question).
The proof of the theorem consists of two steps.
One, showing that if we deform metrics of negative
curvature keeping the spectrum of the laplacian constant
the lengths of closed geodesics are kept constant,
and then showing that this implies that the deformations are isometric.
The later is purely a problem in Riemannian geometry.
Another investigation of the consequences of keeping the
lengths constant was undertaken in [CEG] who showed that, for surfaces
of constant negative curvature, any Hamiltonian deformations that kept
some action invariants had to be smooth canonical transformations.
This result was generalized in [LMM] who showed the same
result is true for Hamiltonian Anosov flows in any dimension.
The methods of this paper
were extended in [L1] [LM] [L2] to show that,
for two dimensional Anosov diffeomorphisms
or three dimensional Anosov flows,
the eigenvalues at periodic orbits
--- which are obviously invariants of $C^1$equivalence ---
form a complete set of invariants for $C^\infty$
or $C^\omega$ equivalence.
A different proof using the theory of SRB measures was constructed in
[Po] and generalized in [L3] to the case of
partially hyperbolic systems.
So it seems that the fact that there is complete set of local
invariants of peridic orbits for smooth conjugacy of Anosov systems is false
in higher dimensions.
Nevertheless,it is true that in some cases the eigenvalues at
periodic orbits are complete sets of invariants for $C^{\tilde k}$
conjugacy and that $C^{k^*}$ conjugacies are $C^\infty$
or $C^\omega$. Unfortunately, ${\tilde k} < k^*$
so that there is a range of
regularities without counterexamples or theorems.
Related to the geodesic flows,
the horocycle flows --- which have no periodic orbits ---
have been shown to be extremely rigid.
In that case, measure theoretic equivalence can be bootstrapped
to differentiable equivalence and, by the
previous results to smooth equivalence.
See for example [FO] and the references there
to previous work of a more algrbraic nature.
Another set of problems which I believe is strongly related
to the previous ones is to show that Cantor sets generated
by dynamical processes are differentiably equivalent if
they are topologically equivalent.
For the Feigenbaum Cantor set, there are several results already
[Ra] [Pa][Su] that show that the topology of the Cantor sets and
the fact that they are dynamically generated forces them to be
$C^1$ equivalent.
Other problems with a similar flavor
are the study of diffeomorphisms that only commute only
with their powers
--i.e., diffeomorphisms without any differentiable symmetry.--
( See [PY1],[PY2] for theorems that show diffeomorphisms with this
property and [PM] for examples of germs of diffeomorphisms with
non trivial centralizer)
or the study of hyperbolic diffeomorphisms
with differentiable stable and unstable
foliations. (See [HK], [Ghy].)
It is sometimes possible to characterize
the systems for which inequalities among dynamical quantities
or characteristic classes are saturated [HK] [KKW].
An interesting generalization of dynamical systems
are actions of groups on manifolds.
On the one hand, they can be considered as non-linear
generalizations of representations and on the
other hand as dynamical systems with a more complicated time
--- a point of view emphasized by the thermodynamic
formalism approach to
dynamical systems. There has been very active program (e.g. [Z]
and references there)
based on the first point of view
to show that actions of {\sl ``large''} groups are
rigid in the sense that, by a smooth change of variables
they can be reduced to a
canonical one.
The dynamical systems point of view has been
exploited in [Hu] to obtain results of rigidity of
$SL(n,{\bf Z})$ on $T^n$.
Coming back to the original problems posed by the papers [GK1] [GK2]
there has been considerable progress made in two papers
that are infuriatingly devoid of heavy machinery and depend only on
stark cleverness. In [Sn] there is an extremely simple machinery
to produce manifolds which are isospectral but not isometric.
In [O] there is a very clever proof that knowledge of the lengths
of closed geodesics and their homotopy classes
determines the manifold up to isometry.
For all that we know [Sn] provides the only mechanism
to produce isospectral but not isometric manifolds
and for a while it was discussed whether it should be
possible to show that isospectral two dimensional manifolds are
related as in [Sn].
I would be embarrassed not to mention the new developments in
symplectic geometry that
are related to symplectic capacities.
Symplectic capacities
are invariants of symplectic maps.
The most transparent constructions of these invariants is
uses Floer cohomology.
Nevertheless, since Floer cohomology is defined
through the gradient flows of a variational principle,
one can also give a variational definition.
There are also rather simple axiomatic characterizations.
Symplectic capacities have many applications.
A tentative one that jumps into mind is that,
for the study of pseudodifferential operators via
the uncertainty principle, one has to pack some regions of phase
space with symplectic images of rectangles.
The theory of symplectic capacities places severe
constraints on what these
images could be. Can this be used to prove bounds
for pseudodifferential operators?.
More firmly,
the theory of symplectic capacities has produced a
proof of the celebrated result that the group of
$C^\infty$ symplectic diffeomorphisms is
$C^0$ closed in the group of diffeomorphisms.
Very readable surveys
with references are the papers
by Viterbo and Hofer in [Mi].
\titlea{}{K.~A.~M. Theory}
\titleb{}{Use of frequencies as parameters.}
The first theorem of K.~A.~M. theory had two very different
proofs. The first one, proposed by Kolmogorov
(see e.g [Ba] for a nodern exposition of the method)
consisted in
selecting one frequency and producing a torus
though several canonical transformations chosen to exhibit that
tori of this frequency indeed existed. The other strategy,
proposed and carried out in great detail
in [Ar2] consisted in studying the resonance regions
and performing the transformations
suggested by first order perturbation theory
in the regions where this transformations
were well defined.
The first method of proof is simpler and became the
dominant method in the
mathematical literature. Especially since it was discovered that
it allowed smoothing schemes that were much sharper than the
simple truncating of Fourier series required by Arnol'd's method
and hence yielded superior results with respect to the
differentiability assumptions required or the differentiability
of the conclusions.
The method could also be abstracted into
implicit function theorems, which could be applied in a variety of
situations.
One of the consequences of this point of view is
that, since the formulation of
implicit function theorems requires one to keep the frequency fixed and
each step of the perturbation changes the frequencies,
it becomes necessary to have extra parameters to adjust
so that the frequency is restored. Hence, it became
customary, when trying to apply a theorem to try to
count free parameters parameters to see if one had enough
to generate counterterms that
allowed the application of thw implicit function theorem.
In the recent mathematical physics literature, the situation was
a little bit more divided due, in good part to the
very nice exposition of Arnol'd's method [CG].
Also, there were theorems such as [FSW] in which the
role of adjusting extra parameters had to be played by performing
probability estimates in a measure space.
An important motivation for overcoming the
constraints imposed by parameter counting is applications to
infinite dimensions, that we will discuss in the next paragraph.
A very direct attack on a problem that
parameter counting said was impossible was [El],
who studied the problem of the preservation of lower dimensional
tori. The method used was to try to change the
frequency considered at every step of the iteration.
Soon afterwards it was remarked that
similar results could be understood as applying
Arnold's method [P\"o].
Then, there has been a flurry of activity using this type of ideas.
Many problems that had been put on the backburner because
the parameter counting said they were impossible became
accessible. One application that I particularly like is the
existence of two dimensional invariant tori in
three dimensional volume preserving flows.
Flows such as those to be considered
in the theorem
can be produced in fluid mechanics
experiments and the existence of the tori
has important physical meaning such as the lack of mixing.
Moreover the tori can be seen by injecting dye!
Such experiments have been performed in the
Center for Nonlinear Dynamics at U.~Texas.
Before that, there were numerical experiments
[Su] confirmed and extended in
[FKP]. Proofs of the theorem were finally obtained
in [ChSu], [DL].
Other theorems in which the frequencies play the
role of extra parameters is [JS].
I certainly expect that there will be many more
as Arnol'd's method of proof becomes better known.
Incidentally, another advantage of using Arnol'd's method
is that the non-degeneracy conditions one obtains are much better
than those required using an implicit function theorem
type of proof. Unfortunately, besides making more difficult
to obtain sharp results with respect to differentiability
it also makes it more difficult to obtain quantitative versions of the
result or to use it to validate numerical computations.
\titleb{}{K.A.M. theory for infinite systems.}
For several years, there has been an active program to study
the thermodynamic limit of K.A.M. theorem.
One aspect of this program was to show that
a system of $N$ particles, each of them with
$d$ degrees of freedom and interacting though short range
forces has invariant tori of positive measure for
values of the perturbation that decrease
like $O( N^{-\alpha})$ [W1], [V].
Moreover in [W3], [W4], there is an analogue of Nekhoroshev
upper bound when the number of degrees of freedom
increases.
In this program, one has to use the
short range of the interaction to show that the
system is effectively almost finite dimensional. This
gives a precise meaning to the physicists intuition that degrees
of freedom are frozen and that the interactions do not count.
Perhaps the frist result for infinitely many
degrees of freedomn is [FSW].
Nevertheless, the system is such that the amplitude of
the oscillations in the degrees of freedom
decreases very fast with the distance to a center.
In effect, the oscillations are localized -- in the
sense of the word in solid state physics.
An abstract version of this result is in [P\"o2].
It is interesting to realize the technical similarities of this
with the series of papers [BS] [PS] in which the opposite is attempted,
namely, to develop a hyperbolic perturbation theory for
hyperbolic systems each of which is hyperbolic
but which are coupled through a short range interaction.
Another interesting development is the application of
methods of K.A.M.\ theory to partial differential equations.
This is not just a case of taking the thermodynamic limit.
If one did that, one would obtain tori of as many dimensions
as the number of degrees of freedom i.e., infinite.
What does it mean to have a function with
infinitely many frequencies? It is not a
very interesting result. There are two interesting papers.
One is the paper [W2] in which the method used
is very similar to Arnol'd's method
applied by defining a measure in the
set of potentials entering in the definition of the
problem. In another set of papers, [K], [K2]
performs an iteration in which the
frequency changes at every stage, but one obtains the
control by direct methods.
\titleb{}{Quantitative versions of the K.A.M.\ theorem.}
From the beginning of K.A.M.~theory it has been a point
of friction with the physicists
that the values for which the theory applies are quite hard to
ascertain rigorously [Mo]. While the mathematicians
were using as the main figure of merit the differentiability properties,
physicists were emphasizing numbers.
In some particular examples, it was possible to obtain reasonable values
[L4] [He] --- incidentally, if one chases though the constants in [Br], one
gets better numbers than those in [L4] as was pointed out
to me by M. Herman. Nevertheless, it is clear that the
methods are quite specific.
One different method was afforded by the use of computer assisted proofs.
The method consists in getting computers to perform
calculations that verify the conditions of a theorem.
The calculations should be performed
taking care to prevent the computers
from using approximations as they do
since they were abandoned by the mathematicians to the engineers.
(See [La] for a review.)
There are two basic strategies. One [Ce], [CC1], [CC2], [CCF] considers
perturbation expansions using methods similar to the
Linstedt series and estimates the remainders and applies
K.A.M.\ theory to them. This has several advantages.
First, it produces results which are valid in open ranges.
Second, the analysis is somewhat easier since the
analytical theorem to be proved are
proved for perturbations of the identity.
The other method, [R], [LR] consists in proving an
implicit function theorem that shows that, if one has an
approximate solution that verifies some
extra hypothesis, then, there is a true one nearby.
Secondly, one has to verify the hypothesis, again with a computer.
This method has the advantage that it requires less calculations
and is insensitive to singularities in the complex domain that,
even if they do not affect the true results, could affect the
convergence of expansions. Indeed, in several cases, there are proofs
that these methods --- up to the limits of numerical analysis ---
will not miss any solution.
Perhaps the main inconvenience of the method as
applied so far is that the
programs developed require severe modifications
in order to be adapted
to other problems. (Nevertheless, the programs in
[R] have been used in four different projects.)
It is to be hoped that, with the new developments
in software tools, soon there will be packages that
not only meet a
new version of Salam's criterion for renormalization theories
but which become routinely used.
The method of computer assisted proofs can also be used to produce upper
bounds of values for which the theorem cannot be true. The first such
paper is [MP]. [Ju] produces a very clever method which is not only
easy to implement but also gets very close to the presumed value. In
[Mu] [MMS] there is a generalization of the method of [MP] to higher
dimensions.
The results of K.A.M.\ theory are nicely counterpointed with the
mathematical results of Aubry-Mather thoery and with the very delicate
numerical work based on renormalization group ideas. Both these subjects
would merit more detailed discussion.
Other mathematical papers that construct systematically examples which
fail to exhibit the conclusions of K.A.M. theorems are
[He2] -- whose methods were used in [NMS]--, [Yo1], [Yo2]
\begref{References}{[MMcG]}
\refmark{[ACKR]} Amik, C., King, E.C.S., Kadanoff, L.P., Rom-Kedar, V.:
Beyond all orders: singular perturbations in a mapping.
\refmark{[An]} Angenent, S.:
Lecture given at IMA conference on twist mappings.
To appear
\refmark{[A]} Anosov, D.V.:
Geodesic flows on closed riemannian manifolds of negative curvature.
Proc. Stekelov Insti. {\bf90} (1967)
\refmark{[AANS]} Anosov, D.V., Arnol'd, V.I., Novikov, S.P., Sinai, V.G.
(eds.):
Dynamical systems I-IV (Encyclopedia of Mathematical Sciences).
Springer-Verlag, 1988
\refmark{[Ar2} Arnol'd, V.I.:
Instability of dynamical systems with several degrees of freedom.
Sov. Mat. Dok. {\bf 5} (1964) 581--585
\refmark{[Ar2} Arnol'd, V.I.:
Proof of a theorm of A.N.~Kolmogorov on the invariance of quasi-periodic
motions under small perturbations of the hamiltonian.
Russ. Math. Surv. {\bf18} (1963) 9--36
\refmark{[BGG]} Benettin, G., Galgani, L., Giorgilli, A.:
A proof of Nelshoroshev's theorem for the stability times in
nearly integrable systems.
\refmark{[BS]} Bunimovich L.A., Sinai, Y.G.:
Space time chaos in coupled map lattices.
Nonlin. {\bf 1} (1988) 491-516
\refmark{[Bar]} Barrar, R.:
Convergence of the VonZeipel procedure.
Cel. Mech. {\bf 2} (1970) 491-504
\refmark{[Br]} Brjno, A.D.:
Analytical form of differential equations.
I. Trans. Mosc. Math. Soc. {\bf25} (1971) 131--288.
II. Ibid. {\bf26} (1972) 199--239
\refmark{[Bu]} Bunimovitch S.:
On the ergodic porperties of nowhere dispersing billiards.
Comm. Math. Phys. {\bf 65} (1979) 295--312
\refmark{[CC1]}Celletti, A.,Chierchia, L.:
Construction of analytic KAM surfaces and effective stability bounds
Comm. Math. Phys. {\bf 118} (1988) 119--161
\refmark{[CC2]}Celletti, A.,Chierchia, L.:
Invariant curves for area-preserving twist maps far from integrable
Jour. Stat. Phys. to appear.
\refmark{[Ce]} Celletti, A.:
Analysis of resonances in the spin orbit problem in celestial mechanics.
Part I., Zamp. {\bf41} (1990) 174--204.
Part II., Ibid. 453--479
\refmark{[ChSu]} Cheng, Chong-quing, Sun, Yi-Sui:
Existence of invariant tori in three-dimesional measure preserving maps.
Cel. Mech. Dyn. Ast. {\bf47} (1990) 275--292
\refmark{[CG]} Chierchia, L., Gallavotti, G.:
Smooth prime integrals for quasi-integrable hamiltonian systems.
Nuov. Cim. {\bf B67} (1982) 277--295
\refmark{[CEG]} Collet, P., Epstein, H., Gallavotti, C.:
Perturbations of geodesic flows on surfaces of constant negative
curvature and their mixing properties.
Comm. Math. Phys. {\bf95} (1984) 61--112
\refmark{[DLT]} Delft, P., Li, L.-C., Tomei, C.:
The Bidiagonal Singular Value Decomposition and Hamiltonian Mechanics.
\refmark{[DL]} Delshams, A., de la Llave, R.:
Existence of quasi-periodic orbits and absence of transport
for volume preserving transformation and flows.
Preprint
\refmark{[De]} Devaney, R.:
Singularities in classical mechanical systems.
In: Ergodic Theory and Dynamical Systems, I.
Birkhauser, Boston, 1981
\refmark{[Don]} Donnay, V.:
Geodesic flow on the two sphere, Part I, positive topological entropy.
Erg. Th. Dyn. Syst. {\bf8} (1988) 531--553
\refmark{[DoL]} Donnay, V., Liverani, C.:
Potentials on the two torus for which the Hamiltonian flow is ergodic.
Comm. Math. Phys. {\bf135} (1991) 367--303
\refmark{[Do]} Douady, R.:
Stabilit\'e ou Instabilit\'e des points fixes elliptiques.
Ann. Sc. ENS {\bf21} (1988) 1--46
\refmark{[El]} Eliasson, L.H.:
Perturbations of stable invariant tori for Hamiltonian systems.
Pub. Sc. Norm. Piser {\bf15} (1988) 115--147
\refmark{[FKP]} Feingold, M., Kadanoff, L.P., Piro, O.:
Passive scalars, three dimensional volume preserving maps and chaos.
J. Stat. Phys. {\bf50} (1988) 529--565
\refmark{[FO]} Feldman, J., Ornstein, D.:
Semi-rigidity of horocycle flows over compact surfaces of variable
negative curvature.
Erg. Th. Dyn. Sys. {\bf7} (1987) 49--72
\refmark{[Fo]} Fomenko, A.:
The symplectic topology of completely integrable Hamiltonian systems.
Russ. Math. Surv. {\bf44} (1989)
\refmark{[FS]} Fontich, E., Sim\'o, C.:
The splitting of separatices for analytic diffeomorphisms.
Erg. Th. Dyn. Syst. {\bf10} (1990) 295--318
\refmark{[FSW]} Fr\"olich, J., Spencer, T., Wayne, C.E.:
Localization in disordered non-linear dynamical systems.
J.~Stat. Phys. {\bf42} (1986) 247--274
\refmark{[Ge]} Gever, J.L.:
The existence of pseudocollisions in the plane.
To appear Jour. Diff. Eq.
\refmark{[Gh]} Ghys, E.:
Flot's d'Anosov dont les feulletages stables sont diff\'erentiables.
Ann. Sci. ENS {\bf20} (1987) 251--271
\refmark{[GK1]} Guillemin, V., Kazhdan, D.:
Some inverse spectral results for negatively curved 2-manifolds.
Topology {\bf19} (1980) 301--312
\refmark{[GK2]} Guillemin, V., Kazhdan, D.:
Some inverse spectral results for negatively curved $n$ manifolds.
AMS Proc. Symp. Pure Math. {\bf35} (1982) 153--180
\refmark{[Gu]} Gutkin, E.:
Asymptotic velocities in classical mechanics.
J. Math. Phys. {\bf30} (1989) 1245--1249
\refmark{[Ha]} Hagedorn, G.A.:
Proof of the Landau-Zenner formula in an adiabatic limit with
small eigenvalue gaps.
Comm. Math. Phys. {\bf136} (1991) 433--451
\refmark{[He]} Hedlund, G.:
The dynamics of geodesic flows.
Bull. Amer. Math. Soc. {\bf45} (1969) 241--260
\refmark{[He]} Herman, M.R.:
Sur les courbes invariantes par les diffeomorphismes de l'anneau.
I. Asterisque (1983) 103--104.
II. Ibid. (1986) 144
\refmark{[He2]} Herman, M.R.:
In\'egalit\'es a priori pour des tores Lagrangunes invariants par
des diffeomorphisms symplectiques.
Pub. Math. IHES {\bf63} (1990) 47--101
\refmark{[Ho]} Hopf, E.:
Statistik der geod\"atischen Linien in Mannigfaltigkesten
negativer Kr\"ummung.
Akad. Wiss. Leipzig {\bf91} (1939) 261--304
\refmark{[Hu]} Hubacher, A.:
Classical scattering theory in 1 dimension.
Preprint
\refmark{[Hu]} Hurder, S.:
Rigiditity for Anosov actions of higher rank lattices.
Preprint
\refmark{[HK]} Hurder, S., Katok, A.:
Differentiability, rigidity and Godbillon-Vey classes for Anosov flows.
To appear Pub. Mat. IHES
\refmark{[JS]} Jak{\u s}i\'c, V., Segert, J.:
Exponential approach to the limit and the Landau-Zenner formula.
Preprint
\refmark{[JS]} Jorba, A., Sim\'o, C.:
On the reducibility of linear differential equations with quasi-periodic
coefficients. Preprint
\refmark{[Ju]} Jungreis, I.:
Ergod. Th. \& Dyn. Syst. {\bf11} (1991) 79--84
\refmark{[KKW]} Katok, A., Knieper, G., Weiss, H.:
Formulas for the derivative and critical points of topological
entropy for Anosov and geodesic flows.
Comm. Math. Phys. {\bf130} (1991) 19--33
\refmark{[KSS]} Kr\'amli, A., Sim\'anyi, N., Sz\'asz, D.:
A ``transversal'' fundamental theorem for semi dispersing billiards.
Comm. Math. Phys. {\bf129} (1949) 535--560.
Erratum Comm. Math. Phys. {\bf138} (1991) 207
\refmark{[Koz]} Kozlov, V.V.:
Integrability nad non-integrability in Hamiltonian mechanics.
Russ. Math. Surv. {\bf 38} (1983) 1-76
\refmark{[Ku]} Kuksin, S.B.:
Perturbation theory for quasiperiodic solutions of infinite dimensional
hamiltonian systems and its applications to the Korteweg-deVries equation.
Math. U.S.S.R. Sb. {\bf64} (1989) 397--413
\refmark{[Ku2]} Kuksin, S.B.:
Perturbation theory for quasi-periodic solutions of infinite dimensional
Hamiltonian systems, Parts I, II, III.
Max Plank preprints
\refmark{[La]} Lanford, O.E.:
Computer assisted proofs in Analysis.
Physica {\bf124A} (1984) 465--470
\refmark{[Lo]} Lochak, P.:
Stabilit\'e en temps exponentiels des syst\`emes hamiltoniens
proches de syst\`emes integrables: Resonances et orbites ferm\'ees.
ENS preprint
\refmark{[L1]} de la Llave, R.:
Invariants for smooth conjugacy of hyperbolic dynamical systems II.
Comm. Math. Phys. {\bf109} (1987) 369--378
\refmark{[L2]} de la Llave, R.:
Analytic regularity of solutions of Livsic's cohomology equation
and application to smooth conjugacy of hyperbolic dynamical systems.
To appear Ergod. Th. Dyn. Syst.
\refmark{[L3]} de la Llave, R.:
Smooth conjugacy and SRB measures for uniformly and non-uniformly
hyperbolic systems.
To appear Comm. Math. Phys.
\refmark{[L4]} de la Llave, R.:
A simple proof of a particular case of $C$ Siegel's center theorem.
Jour. Math. Phys. {\bf24} (1983) 2118--2121
\refmark{[LMM]} de la Llave, R., Marco, J.M., Moriy\'on, R.:
Canonical perturbation theory for Anosov systems and regularity
properties of Livsic's cohomology equation.
Ann. of Math. {\bf123} (1986) 537--611
\refmark{[LM]} de la Llave, R., Moriy\'on, R.:
Invariants for smooth conjugacy of hyperbolic dynamical systems IV.
Comm. Math. Phys. {\bf116} (1988) 185--192
\refmark{[LR]} de la Llave, R., Rana D.:
Accurate strategies for K.A.M. bounds and their implementation.
in Computer aided Proofs in Analysis, Meyer, K., Schmidt, D. (eds) Springer Verlag (1991)
\refmark{[LW]} de la Llave, R., Wayne, C.E.:
Whiskered and low dimensional tori in nearly integrable
dynamical systems. Preprint
\refmark{[MMcG]} Mather, J., McGehee, R.G.:
Solutions of the four body problem which become unbounded in finite time.
Lecture Notes in Physics {\bf38} (1975)
\refmark{[MMS]} McKay R.S., Meiss J.D., Stark, J.:
Converse KAM theory fro symplectic twist maps.
Nonlin. {\bf 2} (1989) 555-570
\refmark{[MP]} McKay, R.S., Percival, I.C.:
Converse K.A.M.\ theory and practice.
Comm. Math. Phys. {\bf98} (1985) 569--512
\refmark{[Mi]} Milojevic, P.S. (ed):
Nonlinear functional analysis.
Marcel Decker, New York, 1990
\refmark{[Mel1]} Melnikov, V.K.:
On some cases of conservation of conditionally periodic motions under small
changes of the Hamiltonian function.
Sov. Mat. Dok. {\bf6} (1965) 1592--1596
\refmark{[Mel2]} Melnikov, V.K.:
A family of conditionally periodic solutions of a Hamiltonian systems.
Sov. Mat. Dok. {\bf9} (1968) 882--885
\refmark{[Mo]} Moser, J.:
Is the solar system stable?
Math. Intell. {\bf1} (1978) 65--71
\refmark{[Mor]} Morales, J.J.:
T\'ecnicas algebraicas para el estudio de la integrabilidad algebraica de sistemas hamiltonianos.
Tesis U. Barcelona (1989)
\refmark{[Mu]} Muldoon, M.:
Gohsts of order on the frontier of chaos.
PhD thesis, Caltech, 1990
\refmark{[Nek]} Nehoroshev, A.N.:
An exponential estimate for the time of stability of nearly
integrable Hamiltonian systems.
Russ. Math. Surv. {\bf32}, no.6 (1977) 1--65
\refmark{[Ne2]} Neishdtadt, A.:
The separation of motions in systems with a rapidly rotating phase.
P.M.M. USSR {\bf40} (1984) 133--139
\refmark{[Ni]} Nikolenko, N.V.:
Invariant asymptotically stable tori of the perturbed Korteweg-deVries
equation.
Russ. Math. Surv. {\bf35} (1980) 139--207
\refmark{[OW]} Ornstein, D., Weiss, B.:
Entropy and isomorphism theorems for actions of amenable groups.
J. Anal. Math. {\bf48} (1987) 1--140
\refmark{[O]} Otal, J.P.:
Le spectre marqu\'e des longeurs des surfaces \`a courbure negative.
Ann. of Math. {\bf131} (1990) 151--162
\refmark{[PY1]} Palis, J., Yoccoz, J-C.:
Rigidity of centralizers of diffeomorphisms.
Ann. Sci ENS {\bf22} (1989) 81--99
\refmark{[PY2]} Palis, J., Yoccoz, J-C.:
Centralizers of Anosov diffeomorphisms on tori.
Ann. Sci ENS {\bf22} (1989) 99--109
\refmark{[PM]} Perez Marco, R.:
Centralizers I: Uncountable centralizers for non linearizable germs of (C,0)
I.H.E.S. preprint 91/23
\refmark{[PS]} Pesin, Y., Sinai, Y.G.
Space time chaos in the system of weakly interactive hyperbolic systems.
Preprint.
\refmark{[Pa]} Paluba, W.:
The Lipschitz condition for the conjugacies of Feigenbaum-like mappings.
Fundamenta Mathematicae {\bf132} (1989) 227--257
\refmark{[Pe]} Pesin, Y.:
Characteristic Lyapunov exponents and smooth ergodic theory.
Russ. Math. Surv. {\bf 32} (1976) 55-114
\refmark{[Pe2]} Pesin, Y.:
An example of a non-ergodic flow with non-zero characteristic exponents.
Funct. Anal. and Appl. {\bf8} (1974) 263--264
\refmark{[Po]} Pollicott, M.:
$C^k$ rigidity for hyperbolic flows.
To appear Israel Jour. Math.
\refmark{[P\"o]} P\"oschel, J.:
On elliptic lower dimensional tori in hamiltonian systems.
Math. Z. {\bf202} (1989) 559--608
\refmark{[P\"o]} P\"oschel, J.:
Small divisors with spatial structure in infinite dimensional dynamical systems.
Comm. Math. Phys. {\bf 127} (1990) 351-393
\refmark{[R]}Rana, D.
Proof of accurate upper and lower bounds to stability domains in small denominator problems:
Princeton thesis (1987)
\refmark{[RGB]} Ramani, A., Gramatikos, B., Bountis, T.:
The Panilev\'e Property and singularity analysis of integrable and
non-integrable systems.
Physics Reports {\bf180} (1989)
\refmark{[RS]} Radin, C., Simon, B.:
Invariant domains for the time-dependent Schr\"odinger equation.
Jour. Diff. Eq. {\bf 29} (1978) 289-296
\refmark{[Ra]} Rand, D.:
Global phase space universality, smooth conjugacies and
renormalisations. The $C^{1+\alpha}$ case.
Nonlinearity {\bf1} (1988) 181--202
\refmark{[SM]} Siegel, C.L., Moser, J.:
Lectures on Celestial Mechanics.
Springer-Verlag, 1971
\refmark{[SCh]} Sinai, Y.G., Chernoff, S.:
Ergodic porperties of certain systems of two dimensional discs and three dimensional balls.
Russ. Math. Surv. {\bf 42} (1987) 181-207
\refmark{[Si]} Sinai, Ya.G.:
Dynamical systems with elastic reflections. Ergodic properties
of dispersing billiards.
Russ. Math. Surv. {\bf25} no.2 (1970) 137--189
\refmark{[Su]} Sullivan, D.:
Bounds, quadratic differentials and renormalization conjectures.
Preprint
\refmark{[Sn]} Sunada, T.:
Riemannian coverings and isospectral manifolds.
Ann. of Math. {\bf121} (1985) 169--186
\refmark{[Sun]} Sun, Y.S.:
On the measure preserving mappings with odd dimension.
Celest. Mech. {\bf30} (1983) 7--19
\refmark{[Tr]} Treshev, D.V.:
The mechanisms of destruction of resonance tori of hamiltonian systems.
Math. U.S.S.R. Sbor. {\bf68} (1991) 181--203
\refmark{[V]} Vitott, M.
Th\'eorie classique des perturbations en grand nombre de degres de libert\'e.
Univ. de Marseille, Thesis (1985)
\refmark{[W]} Wojtkowski, M.:
Principles for the design of Billiards with Non-vanishing
Lyapunov exponents.
Comm. Math. Phys. {\bf105} (1986) 391--414
\refmark{[W1} Wayne, C.E.:
The K.A.M. theory of systems with short range interactions.
Part I, Comm. Math. Phys. {\bf 96} (1984) 311-329; Part II ibid. 331--344
\refmark{[W2]}Wayne, C.E.:
Periodic and quasi-periodic solutions of nonlinear wave equations via K.A.M. Theory.
Comm. Math. Phys. {\bf 127} (1990) 479--528
\refmark{[W3]}Wayne, C. E.:
On the elimination of non-resonant harmonics.
Comm. Math. Phys. {\bf 103} (1986) 351-386
\refmark{[W3]}Wayne, C. E.:
Uniform estimates on trajectories in hamiltonian systems with many degrees of freedom.
Comm. Math. Phys. {\bf 104} (1986) 21-36
\refmark{[Xi]} Xia, Z.:
The existence of the non-collision singularities in Newtonian systems.
PhD thesis, Northwestern University, 1988
\refmark{[Y1]}Yoccoz, J-C.:
Th\'eor\`eme de Siegel, polyn\^omes quadratiques et nombres de Brjuno
Preprint
\refmark{[Y2]}Yoccoz, J-C.:
Lin\'earization des germes de diffeomorphismes holomorphes de (C,0)
C.R.Acad.Sci. Paris {\bf 306} (1988) 555-558
\refmark{[Z]} Zimmer, R.:
Infinitesimal rigidity for smooth actions of discrete subgroups
of Lie groups.
\endref
\end