%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This is a plain TeX file. To print, save in a file file.tex %
% and run through normal tex (tested with version 3.141) %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\magnification=\magstep1
%The original version of these macros is due to J.P. Eckmann
%
%\magnification \magstep1
\vsize=22 truecm
\hsize=16 truecm
\hoffset=0.8 truecm
\normalbaselineskip=5.25mm
\baselineskip=5.25mm
\parskip=10pt
\immediate\openout1=key
\font\titlefont=cmbx10 scaled\magstep1
\font\authorfont=cmcsc10
\font\footfont=cmr7
\font\sectionfont=cmbx10 scaled\magstep1
\font\subsectionfont=cmbx10
\font\small=cmr7
\font\smaller=cmr5
\font\mathsmaller=cmsy5
%%%%%constant subscript positions%%%%%
\fontdimen16\tensy=2.7pt
\fontdimen17\tensy=2.7pt
\fontdimen14\tensy=2.7pt
%%%%%%%%%%%%%%%%%%%%%%
%%% macros %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\def\dowrite #1{\immediate\write16 {#1} \immediate\write1 {#1} }
%\headline={\ifnum\pageno>1 {\hss\tenrm-\ \folio\ -\hss} \else {\hfill}\fi}
\newcount\EQNcount \EQNcount=1
\newcount\SECTIONcount \SECTIONcount=0
\newcount\APPENDIXcount \APPENDIXcount=0
\newcount\CLAIMcount \CLAIMcount=1
\newcount\SUBSECTIONcount \SUBSECTIONcount=1
\def\SECTIONHEAD{X}
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
\def\QED{\hfill\smallskip
\line{\hfill\vrule height 1.8ex width 2ex depth +.2ex
\ \ \ \ \ \ }
\bigskip}
% These ones cannot be used in amstex
% Make te symbols just bold
%\def\real{{\bf R}}
\def\rational{{\bf Q}}
%\def\natural{{\bf N}}
%\def\complex{{\bf C}}
%\def\integer{{\bf Z}}
\def\torus{{\bf T}}
% Make the symbols using kerning
\def\natural{{\rm I\kern-.18em N}}
\def\integer{{\rm Z\kern-.32em Z}}
\def\real{{\rm I\kern-.2em R}}
\def\complex{\kern.1em{\raise.47ex\hbox{
$\scriptscriptstyle |$}}\kern-.40em{\rm C}}
%
% These ones can only be used in amstex
%
%\def\real{{\Bbb R}}
%\def\rational{{\Bbb Q}}
%\def\natural{{\Bbb N}}
%\def\complex{{\Bbb C}}
%\def\integer{{\Bbb Z}}
%\def\torus{{\Bbb T}}
%
%
%
\def\Re{{\rm Re\,}}
\def\Im{{\rm Im\,}}
\def\PROOF{\medskip\noindent{\bf Proof.\ }}
\def\REMARK{\medskip\noindent{\bf Remark.\ }}
\def\NOTATION{\medskip\noindent{\bf Notation.\ }}
\def\PRUEBA{\medskip\noindent{\bf Demostraci\'on.\ }}
\def\NOTA{\medskip\noindent{\bf Nota.\ }}
\def\NOTACION{\medskip\noindent{\bf Notaci\'on.\ }}
\def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax}
\def\equ(#1){\ifundefined{e#1}$\spadesuit$#1 \dowrite{undefined equation #1}
\else\csname e#1\endcsname\fi}
\def\clm(#1){\ifundefined{c#1}$\clubsuit$#1 \dowrite{undefined claim #1}
\else\csname c#1\endcsname\fi}
\def\EQ(#1){\leqno\JPtag(#1)}
\def\NR(#1){&\JPtag(#1)\cr} %the same as &\tag(xx)\cr in eqalignno
\def\JPtag(#1){(\SECTIONHEAD.
\number\EQNcount)
\expandafter\xdef\csname
e#1\endcsname{(\SECTIONHEAD.\number\EQNcount)}
\dowrite{ EQ \equ(#1):#1 }
\global\advance\EQNcount by 1
}
\def\CLAIM #1(#2) #3\par{
\vskip.1in\medbreak\noindent
{\bf #1~\SECTIONHEAD.\number\CLAIMcount.} {\sl #3}\par
\expandafter\xdef\csname c#2\endcsname{#1\
\SECTIONHEAD.\number\CLAIMcount}
%\immediate \write16{ CLAIM #1 (\number\SECTIONcount.\number\CLAIMcount) :#2}
%\immediate \write1{ CLAIM #1 (\number\SECTIONcount.\number\CLAIMcount) :#2}
\dowrite{ CLAIM #1 (\SECTIONHEAD.\number\CLAIMcount) :#2}
\global\advance\CLAIMcount by 1
\ifdim\lastskip<\medskipamount
\removelastskip\penalty55\medskip\fi}
\def\CLAIMNONR #1(#2) #3\par{
\vskip.1in\medbreak\noindent
{\bf #1~#2} {\sl #3}\par
\global\advance\CLAIMcount by 1
\ifdim\lastskip<\medskipamount
\removelastskip\penalty55\medskip\fi}
\def\SECTION#1\par{\vskip0pt plus.3\vsize\penalty-75
\vskip0pt plus -.3\vsize\bigskip\bigskip
\global\advance\SECTIONcount by 1
\def\SECTIONHEAD{\number\SECTIONcount}
\immediate\dowrite{ SECTION \SECTIONHEAD:#1}\leftline
%{{\sectionfont }\ {\sectionfont #1} }
{\sectionfont \SECTIONHEAD.\ #1 }
\EQNcount=1
\CLAIMcount=1
\SUBSECTIONcount=1
\nobreak\smallskip\noindent}
\def\APPENDIX#1\par{\vskip0pt plus.3\vsize\penalty-75
\vskip0pt plus -.3\vsize\bigskip\bigskip
\def\SECTIONHEAD{\ifcase \number\APPENDIXcount X\or A\or B\or C\or D\or E\or F \fi}
\global\advance\APPENDIXcount by 1
\vfill \eject
\immediate\dowrite{ APPENDIX \SECTIONHEAD:#1}\leftline
{\titlefont APPENDIX \SECTIONHEAD: }
{\sectionfont #1}
\EQNcount=1
\CLAIMcount=1
\SUBSECTIONcount=1
\nobreak\smallskip\noindent}
\def\SECTIONNONR#1\par{\vskip0pt plus.3\vsize\penalty-75
\vskip0pt plus -.3\vsize\bigskip\bigskip
\global\advance\SECTIONcount by 1
\immediate\dowrite{SECTION:#1}\leftline
{\sectionfont #1}
\EQNcount=1
\CLAIMcount=1
\SUBSECTIONcount=1
\nobreak\smallskip\noindent}
\def\SUBSECTION#1\par{\vskip0pt plus.2\vsize\penalty-75
\vskip0pt plus -.2\vsize\bigskip\bigskip
\def\SUBSECTIONHEAD{\number\SUBSECTIONcount}
\immediate\dowrite{ SUBSECTION \SECTIONHEAD.\SUBSECTIONHEAD :#1}\leftline
{\subsectionfont
\SECTIONHEAD.\number\SUBSECTIONcount.\ #1}
\global\advance\SUBSECTIONcount by 1
\nobreak\smallskip\noindent}
\def\SUBSECTIONNONR#1\par{\vskip0pt plus.2\vsize\penalty-75
\vskip0pt plus -.2\vsize\bigskip\bigskip
\immediate\dowrite{SUBSECTION:#1}\leftline{\subsectionfont
#1}
\nobreak\smallskip\noindent}
%%%%%%%%%%%%%TITLE PAGE%%%%%%%%%%%%%%%%%%%%
\let\endarg=\par
\def\finish{\def\endarg{\par\endgroup}}
\def\start{\endarg\begingroup}
\def\getNORMAL#1{{#1}}
\def\TITLE{\beginTITLE\getTITLE}
\def\beginTITLE{\start
\titlefont\baselineskip=1.728
\normalbaselineskip\rightskip=0pt plus1fil
\noindent
\def\endarg{\par\vskip.35in\endgroup}}
\def\getTITLE{\getNORMAL}
\def\AUTHOR{\beginAUTHOR\getAUTHOR}
\def\beginAUTHOR{\start
\vskip .25in\rm\noindent\finish}
\def\getAUTHOR{\getNORMAL}
\def\FROM{\beginFROM\getFROM}
\def\beginFROM{\start\baselineskip=3.0mm\normalbaselineskip=3.0mm
\obeylines\sl\finish}
\def\getFROM{\getNORMAL}
\def\ENDTITLE{\endarg}
\def\ABSTRACT#1\par{
\vskip 1in {\noindent\sectionfont Abstract.} #1 \par}
\def\ENDABSTRACT{\vfill\break}
\def\TODAY{\number\day~\ifcase\month\or January \or February \or March \or
April \or May \or June
\or July \or August \or September \or October \or November \or December \fi
\number\year}
\newcount\timecount
\timecount=\number\time
\divide\timecount by 60
\def\DRAFT{\font\footfont=cmti7
\footline={{\footfont \hfil File:\jobname, \TODAY, \number\timecount h}}
}
%%%%%%%%%%%%%%%%BIBLIOGRAPHY%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\period{\unskip.\spacefactor3000 { }}
%
% ...invisible stuff
%
\newbox\noboxJPE
\newbox\byboxJPE
\newbox\paperboxJPE
\newbox\yrboxJPE
\newbox\jourboxJPE
\newbox\pagesboxJPE
\newbox\volboxJPE
\newbox\preprintboxJPE
\newbox\toappearboxJPE
\newbox\bookboxJPE
\newbox\bybookboxJPE
\newbox\publisherboxJPE
\def\refclearJPE{
\setbox\noboxJPE=\null \gdef\isnoJPE{F}
\setbox\byboxJPE=\null \gdef\isbyJPE{F}
\setbox\paperboxJPE=\null \gdef\ispaperJPE{F}
\setbox\yrboxJPE=\null \gdef\isyrJPE{F}
\setbox\jourboxJPE=\null \gdef\isjourJPE{F}
\setbox\pagesboxJPE=\null \gdef\ispagesJPE{F}
\setbox\volboxJPE=\null \gdef\isvolJPE{F}
\setbox\preprintboxJPE=\null \gdef\ispreprintJPE{F}
\setbox\toappearboxJPE=\null \gdef\istoappearJPE{F}
\setbox\bookboxJPE=\null \gdef\isbookJPE{F} \gdef\isinbookJPE{F}
\setbox\bybookboxJPE=\null \gdef\isbybookJPE{F}
\setbox\publisherboxJPE=\null \gdef\ispublisherJPE{F}
}
\def\ref{\refclearJPE\bgroup}
\def\no {\egroup\gdef\isnoJPE{T}\setbox\noboxJPE=\hbox\bgroup}
\def\by {\egroup\gdef\isbyJPE{T}\setbox\byboxJPE=\hbox\bgroup}
\def\paper{\egroup\gdef\ispaperJPE{T}\setbox\paperboxJPE=\hbox\bgroup}
\def\yr{\egroup\gdef\isyrJPE{T}\setbox\yrboxJPE=\hbox\bgroup}
\def\jour{\egroup\gdef\isjourJPE{T}\setbox\jourboxJPE=\hbox\bgroup}
\def\pages{\egroup\gdef\ispagesJPE{T}\setbox\pagesboxJPE=\hbox\bgroup}
\def\vol{\egroup\gdef\isvolJPE{T}\setbox\volboxJPE=\hbox\bgroup\bf}
\def\preprint{\egroup\gdef
\ispreprintJPE{T}\setbox\preprintboxJPE=\hbox\bgroup}
\def\toappear{\egroup\gdef
\istoappearJPE{T}\setbox\toappearboxJPE=\hbox\bgroup}
\def\book{\egroup\gdef\isbookJPE{T}\setbox\bookboxJPE=\hbox\bgroup\it}
\def\publisher{\egroup\gdef
\ispublisherJPE{T}\setbox\publisherboxJPE=\hbox\bgroup}
\def\inbook{\egroup\gdef\isinbookJPE{T}\setbox\bookboxJPE=\hbox\bgroup\it}
\def\bybook{\egroup\gdef\isbybookJPE{T}\setbox\bybookboxJPE=\hbox\bgroup}
\def\endref{\egroup \sfcode`.=1000
\if T\isnoJPE \item{[\unhbox\noboxJPE\unskip]}
\else \item{} \fi
\if T\isbyJPE \unhbox\byboxJPE\unskip: \fi
\if T\ispaperJPE \unhbox\paperboxJPE\unskip\period \fi
\if T\isbookJPE ``\unhbox\bookboxJPE\unskip''\if T\ispublisherJPE, \else.
\fi\fi
\if T\isinbookJPE In ``\unhbox\bookboxJPE\unskip''\if T\isbybookJPE,
\else\period \fi\fi
\if T\isbybookJPE (\unhbox\bybookboxJPE\unskip)\period \fi
\if T\ispublisherJPE \unhbox\publisherboxJPE\unskip \if T\isjourJPE, \else\if
T\isyrJPE \ \else\period \fi\fi\fi
\if T\istoappearJPE (To appear)\period \fi
\if T\ispreprintJPE Preprint\period \fi
\if T\isjourJPE \unhbox\jourboxJPE\unskip\ \fi
\if T\isvolJPE \unhbox\volboxJPE\unskip, \fi
\if T\ispagesJPE \unhbox\pagesboxJPE\unskip\ \fi
\if T\isyrJPE (\unhbox\yrboxJPE\unskip)\period \fi
}
%*************** TO GET SMALLER FONT FAMILIES *****************
\newskip\ttglue
% ********** EIGHT POINT **************
\def\eightpoint{\def\rm{\fam0\eightrm}
\textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
\textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei
\textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
\textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
\textfont\itfam=\eightit \def\it{\fam\itfam\eightit}
\textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}
\textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}
\textfont\bffam=\eightbf \scriptfont\bffam=\sixbf
\scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}
\tt \ttglue=.5em plus.25em minus.15em
\normalbaselineskip=9pt
\setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}
\let\sc=\sixrm \let\big=\eightbig \normalbaselines\rm}
\font\eightrm=cmr8 \font\sixrm=cmr6 \font\fiverm=cmr5
\font\eighti=cmmi8 \font\sixi=cmmi6 \font\fivei=cmmi5
\font\eightsy=cmsy8 \font\sixsy=cmsy6 \font\fivesy=cmsy5
\font\eightit=cmti8 \font\eightsl=cmsl8 \font\eighttt=cmtt8
\font\eightbf=cmbx8 \font\sixbf=cmbx6 \font\fivebf=cmbx5
\def\eightbig#1{{\hbox{$\textfont0=\ninerm\textfont2=\ninesy
\left#1\vbox to6.5pt{}\right.\enspace$}}}
%************** NINE POINT *****************
\def\ninepoint{\def\rm{\fam0\ninerm}
\textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
\textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei
\textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
\textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
\textfont\itfam=\nineit \def\it{\fam\itfam\nineit}
\textfont\slfam=\ninesl \def\sl{\fam\slfam\ninesl}
\textfont\ttfam=\ninett \def\tt{\fam\ttfam\ninett}
\textfont\bffam=\ninebf \scriptfont\bffam=\sixbf
\scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\ninebf}
\tt \ttglue=.5em plus.25em minus.15em
\normalbaselineskip=11pt
\setbox\strutbox=\hbox{\vrule height8pt depth3pt width0pt}
\let\sc=\sevenrm \let\big=\ninebig \normalbaselines\rm}
\font\ninerm=cmr9 \font\sixrm=cmr6 \font\fiverm=cmr5
\font\ninei=cmmi9 \font\sixi=cmmi6 \font\fivei=cmmi5
\font\ninesy=cmsy9 \font\sixsy=cmsy6 \font\fivesy=cmsy5
\font\nineit=cmti9 \font\ninesl=cmsl9 \font\ninett=cmtt9
\font\ninebf=cmbx9 \font\sixbf=cmbx6 \font\fivebf=cmbx5
\def\ninebig#1{{\hbox{$\textfont0=\tenrm\textfont2=\tensy
\left#1\vbox to7.25pt{}\right.$}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%************* The picture definitions ******************
\def\picture#1#2#3{
\vbox to #3 in{
\hbox to 2 in
{\special{psfile = #1.ps angle = 270 hscale = 40 vscale = 40 }}
\hfil \vfil}
\kern1.3cm
\penalty 10000
\vbox{\eightpoint #2 \hfil}
\vskip0.3truecm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% SOME SMALL TRICKS%%%%%%%%%%
\def \breakline{\vskip 0em}
\def \script{\bf}
\def \norm{\vert \vert}
\def \endnorm{\vert \vert}
\def \cite#1{{[#1]}}
\def\cite#1{{\rm [#1]}}
\def\bref#1{{\rm [~\enspace~]}} % blank ref cite
\def\degree{\mathop{\rm degree}\nolimits}
\def\mod{\mathop{\rm mod}\nolimits}
\def\bT{{\bf T}}
\def\frac#1#2{{#1\over#2}}
\def\Norm{{\|~\enspace~\|}}
\def\runninghead#1#2{\voffset=2\baselineskip\nopagenumbers
\headline={\ifodd\pageno\rightheadline\else
\leftheadline\fi}
\def\rightheadline{{\sl#1}\hfill{\rm\folio}}
\def\leftheadline{{\rm\folio}\hfill{\sl#2}}}
% \DRAFT
\overfullrule=0pt
\def\cite#1{{\rm [#1]}}
\def\bref#1{{\rm [~\enspace~]}} % blank ref cite
\def\degree{\mathop{\rm degree}\nolimits}
\def\mod{\mathop{\rm mod}\nolimits}
\def\bT{{\bf T}}
\def\frac#1#2{{#1\over#2}}
\def\Norm{{\|~\enspace~\|}}
\TITLE
Approximation of invariant surfaces by periodic orbits in high-dimensional
maps. Some rigorous results.
\ENDTITLE
\AUTHOR
Stathis Tompaidis
\footnote{$^1$}{Partial support from NSERC grant OGP-0121848}
\footnote{$^2$}{e-mail: {\tt stathis@lie.univ-rennes1.fr}}
%\footnote{$^2$}{e-mail: {\tt stathis@math.toronto.edu}}
%\footnote{$^3$}{Current address: IRMAR, Univ. de Rennes 1, 35042 Rennes Cedex,
%France}
\FROM
IRMAR
Universit\'e de Rennes I
35042 Rennes Cedex
France
%Department of Mathematics
%University of Toronto
%100 St. George Street
%Toronto, ON M5S 1A1
%Canada
\ENDTITLE
\ABSTRACT
The existence of an invariant surface in high-dimensional systems greatly
influences the behavior in a neighborhood of the invariant surface. We
prove theorems that explain the behavior of periodic orbits in the
vicinity of an invariant surface for symplectic maps and quasi-periodic
perturbations of symplectic maps.
Our results allow for efficient numerical algorithms that can
serve as an indication for the breakdown of invariant surfaces.
\ENDABSTRACT
\runninghead{Approximation by periodic orbits. Rigorous results}{S. Tompaidis}
\SECTION Introduction
Periodic orbits have long served as tools to study the long term behavior
of dynamical systems (as witnessed, for example, by Poincar\'e,
see \cite{Po93}).
In 1979, Greene proposed a numerical criterion, based on the behavior of
periodic orbits, to determine the parameter values at which breakdown
of certain invariant circles of twist maps of the annulus occurs.
The criterion (henceforth
called ``Greene's criterion'', see \cite{Gr79} for a precise formulation)
is remarkably
accurate and has provided valuable intuition that led to the formulation of
a renormalization group theory for the breakdown of invariant circles for
twist maps of the annulus (see \cite{McK82}).
Determining the parameter values at which breakdown of invariant surfaces occurs
has significant practical importance, as invariant surfaces present barriers to
phase-space diffusion. %Even in the case of high-dimensional maps,
%when the
%invariant surfaces are only partial barriers, they are very ``sticky'',
%i.e. orbits stay in their neighborhood for exponentially long times. This
%result, similar to Nekhorosev's estimates for phase space diffusion for
%systems close to integrable (see \cite{Nek??}),
%was proven by Perry and Wiggins (see
%\cite{PW94} where the term ``sticky'' was also introduced).
Part of Greene's criterion (initially conjectured in \cite{Gr79} and
later proved in \cite{McK92, FL92}) asserts that twist maps of the
annulus admit an invariant circle
with diophantine rotation number as long as
a certain limit, taken along periodic orbits in the neighborhood of the
invariant circle and based on their stability,
is equal to zero. Moreover, if the invariant circle is
analytic, the limit is reached exponentially fast. Such behavior can,
and has been, efficiently investigated numerically.
We present a similar result for certain
high-dimensional symplectic and volume-preserving
maps, satisfying non-degeneracy
assumptions. If an invariant surface $\Gamma$
exists and is analytic, or sufficiently differentiable, and motion on
$\Gamma$ is conjugate to rigid rotation with a diophantine rotation vector,
we show that
all the eigenvalues of the derivative of the map along periodic orbits
in a neighborhood of $\Gamma$ tend to $1$ (exponentially, if
the invariant surface is analytic) as the periodic orbit approaches
$\Gamma$. A precise statement is given in section 2.
Our results are of a local nature and involve only a neighborhood of the
invariant surface. Existence of an invariant surface imposes
severe restrictions for the map in a neighborhood of the surface.
We show that in an appropriate neighborhood of the invariant
surface the map
is close to integrable and using a perturbative argument we can control
the behavior of periodic orbits. One can use the distance from the
invariant surface as a smallness parameter and deduce that periodic
orbits with rotation vectors close to the rotation vector of the
invariant surface exist close to the surface. In \cite{PW94} similar
ideas were used to deduce long-term stability for orbits that come
close to an invariant surface.
\SECTION Notation and statement of results
We will study two distinct cases: (a) symplectic maps and
(b) volume-preserving maps that are quasiperiodic perturbations
of symplectic maps.
In the first case we consider $C^r$ and analytic
maps $f$ from the space $\torus^d \times \real^d$ to itself,
satisfying
\item{(i)} they preserve the natural symplectic 2-form
$ \omega = \sum_{i=1}^d d\phi_i \wedge dA_i$
\item{(ii)}
$\partial \phi' / \partial A$ is a non-singular matrix (of dimension $d$)
\noindent where $\phi'$ the first coordinate
of $\tilde f (\phi,A)$ for $\tilde f$ a
lift of $f$. We will call a function $f$ satisfying $(i)$ and
$(ii)$ a $2d$-dimensional non-singular symplectic map. Examples
of $C^r$ maps satisfying $(i), (ii)$ for $d=1$ are called (positive or
negative) twist maps of the
annulus.
In the second case we consider maps
$f: \torus^{d+e} \times \real^d \to \torus^{d+e} \times \real^d$
that are periodic or quasi-periodic skew-products on
$\torus^e$ where
$f|_{\torus^d \times \real^d} : \torus^d \times \real^d \to \torus^d \times
\real^d$
satisfies $(i), (ii)$.% and $f|_{\torus^e} : \torus^e \to \torus^e$
%is rigid rotation by a fixed rotation vector.
%Such a volume-preserving map can be considered as a periodic or
%quasi-periodic perturbation of $f|_{\torus^d \times \real^d}$
%(depending on the fixed rotation vector).
We say that $\bf x$ is a periodic orbit
of type $(P/N)$,
$P\in \integer^c$, where $c=d$ for the case of the
symplectic maps or $c=d+e$ for the case of volume-preserving maps,
$N \in \natural^*(\equiv \natural - \{0\})$, if
$f^N ({\bf x}) = {\bf x}$ and
$\tilde f^N ({\bf \tilde x}) = {\bf \tilde x} + (P, 0)$,
where $\tilde f, \bf \tilde x$ a (fixed) lift of $f, \bf x$ to the universal
cover of $\torus^c \times \real^d$. We will
call $N$ the period of the orbit.
Notice that only periodic skew-products can have periodic orbits.
For $c$-vectors we will use the norm
$\|{\bf \omega}\|_c = \sum_{i=1}^c |\omega_i| $.
We define the {\it rotation vector} of an orbit of $\tilde f$ as the
$c$-dimensional vector
$$ \omega = \lim_{i\to \infty} \frac{\pi_1(\tilde f^i (x,y)) - x}{i} $$
if the limit exists, where $\pi_1$ the projection on the first
$c$ (angle) coordinates $\pi_1 (x,y) = x$. For a periodic orbit of type $(P/N)$
the rotation vector is $\omega = P/N$.
We will consider sets with rotation vectors that are not well approximated by
rational vectors. We define a $c$-dimensional vector to be of type
$(K, \tau)$ if
$$ | P\cdot\omega | \ge \frac{K}{\|P\|_c^\tau}, P \in \integer^c, P\ne 0, K> 0
\EQ(diophantine)$$
It is known (see \cite{Ar88}) that if $\tau > c-1 $ the set of vectors of
type $(K, \tau)$ has positive Lebesgue measure in the unit $c$-dimensional
cube.
We now state our results for periodic
orbits that approach invariant sets of $f$.
\CLAIM{Theorem}(residue_smooth)
Let $f \in C^r(\torus^d\times\real^d), r>1$
satisfy (i), (ii) and admit a $C^r$ invariant
surface $\Gamma$,
homotopic to $\torus^d$,
on which the motion is
$C^r$ conjugate to rigid rotation with
rotation vector ${\bf \omega}$ of type $(K, \tau)$.
Moreover assume that in a neighborhood
of $\Gamma$ there are periodic orbits $x_{(P/N)}$ of type ($P/N$) for
$\|N \omega - P \|_d$ small enough.\hfil \breakline
Then, for $k \in \natural, k < \frac{r-1}{\tau}$
we can find $D_k > 0$, such that
the eigenvalues $\lambda_1, \dots, \lambda_{2d}$
of the derivative $Df^N (x_{(P/N)})$ satisfy
$$|\lambda_i -1 | \le D_k \| N{\omega} - P\|_d^{k/2} N ,\ i = 1,\dots ,2d$$
In the case where the map $f$ and the invariant surface are analytic
in a poly-strip $I_\delta$ around the invariant surface $\Gamma$
%= \{\phi \in \torus^d : |\Im \phi_i| \le \delta \}$
and analytically conjugate to rigid rotation, we can compute the coefficients
$D_k$ and choose the $k$ that gives the best bound.
\CLAIM{Theorem}(residue_anal)
Let $f:\torus^d\times\real^d\to\torus^d\times\real^d$ analytic
satisfy (i), (ii) and admit an analytic invariant
surface $\Gamma$,
homotopic to $\torus^d$,
on which the motion is
analytically conjugate, with conjugacy $\gamma$, to rigid rotation with
rotation vector ${\bf \omega}$ of type $(K, \tau)$.
Moreover assume that in a neighborhood
of $\Gamma$ there are periodic orbits $x_{(P/N)}$
of type ($P/N$) for
$\|N \omega - P \|_d$ small enough.
If $f,\gamma$ are bounded in a neighborhood of the invariant surface
then the eigenvalues $\lambda_1, \dots, \lambda_{2d}$
of the derivative $Df^N (x_{(P/N)})$ satisfy
$$|\lambda_i -1 | \le
\tilde D_1 N
\exp ( -\tilde D_2 \| N\omega - P\|_d^{\frac{-1}{2(1+\tau)}}) $$
where $\tilde D_2$ depends on the width of the domain of
analyticity of $f, \gamma$.
In the case $d=1$, the behavior of the eigenvalues is completely determined
by the trace of the derivative along the periodic orbit.
In analogy with that case, we define the residue of a periodic orbit
with period $N$, as
$$R({\bf x}) = \frac{1}{4d} \left [ 2d - {\rm Tr}(D f^N ({\bf x})) \right ]
\EQ(residue)$$
Our definition is an extension of the one used by Greene in \cite{Gr79}
for two-dimensional twist maps of the
annulus. The factor $(4d)^{-1}$ assures that the residue of elliptic
periodic orbits (i.e. orbits for which the eigenvalues of $D\tilde f^N$ lie
on the unit circle) is between zero and one.
In \cite{Gr79}
Greene formulated a criterion for the breakdown of invariant curves of
twist maps based on the behavior of the residue of periodic orbits. As
indicated by \clm(residue_anal), an
analog of the criterion in higher dimensions should consider the
behavior of {\it additional} quantities, other than the residue, such as
the {\it eigenvalues} of $Df^N$ along periodic orbits.
Notice that, due to invariance under cyclic permutations,
the residue of a periodic orbit is the same for all the points of the orbit.
Also, since the definition only involves derivatives, the residue is invariant
under $C^1$ changes of variables. For integrable maps (i.e. a map
conjugate to $\tilde g (x,y) = (x + h(y), y)$ for $h : \real^d \to \real^d$)
the residue of all periodic orbits is zero. From \clm(residue_smooth),
\clm(residue_anal) we have the following corollary:
\CLAIM{Corollary}(residue_cor)
Let $f \in C^r(\torus^d\times\real^d), r>1$ (analytic)
satisfy (i), (ii) and admit a $C^r$ (analytic) invariant
surface $\Gamma$,
homotopic to $\torus^d$,
on which the motion is
$C^r$ (analytically) conjugate to rigid rotation with
rotation vector ${\bf \omega}$ of type $(K, \tau)$.
Moreover assume that in a neighborhood
of $\Gamma$ there are periodic orbits $x_{(P/N)}$ of type ($P/N$) for
$\|N \omega - P \|_d$ small enough.\hfil \breakline
Then, for $k \in \natural, k < \frac{r-1}{\tau}$
we can find $C_k > 0$, such that
$$|R(x) | \le C_k \| N{\omega} - P\|_d^{k/2} N$$
$$(|R(x) | \le \tilde C_1 N
\exp ( -\tilde C_2 \| N\omega - P\|_d^{\frac{-1}{2(1+\tau)}}) )$$
\REMARK
In the case $d=1$ the continued-fraction convergents to $\omega$
provide a series of numbers $\{ M_i/N_i\}_{i=0}^{\infty}$ such that
$$|\omega -M_i/N_i | \le K N_i^{-2}, {\rm\ for\ all\ } i, \omega
\EQ(contfrac)$$
In that case it is possible to
show that if an analytic invariant curve exists
$$ \lim_{i \to \infty} \sup | R(x_i) |^{1/N_i} \le 1 $$
where the limit is taken along continued fraction convergents.
Unfortunately, in higher dimensions,
we are not aware of an approximation scheme that
can produce convergents to an arbitrary rotation vector with $d$
components that satisfy an inequality similar to \equ(contfrac)
(such schemes exist for certain classes of rotation vectors though --
e.g. golden vectors of the Jacobi-Perron algorithm for $d=2$, see
\cite{Kos91}).
\REMARK
\clm(residue_smooth) and \clm(residue_anal) are local results that
apply in a neighborhood of the invariant surface. Thus, assumptions
$(i)$, $(ii)$ can be relaxed to assumptions $(i)$ and $(ii)$
holding only in a neighborhood of the
invariant surface $\Gamma$.
For the case of volume-preserving maps that are quasi-periodic
skew-products of symplectic maps over $\torus^e$ we have
\CLAIM{Theorem}(volpres)
Let
$f:\torus^{d+e}\times \real^d\to \torus^{d+e}\times \real^d \in C^r, r>1$
(analytic)
be a quasi-periodic skew-product of a symplectic map satisfying
(i), (ii) over $\torus^e$ such that
$f|_{\torus_e}$ is rigid rotation with a diophantine rotation vector.
Assume that $f$ admits a $C^r$ (analytic) invariant
surface $\Gamma$, homotopic to $\torus^{d+e}$,
on which the motion is
$C^r$ (analytically) conjugate to rigid rotation with
rotation vector ${\bf \omega}$ of type $(K, \tau)$.
Moreover assume that in a neighborhood
of $\Gamma, f$ there are periodic orbits $x_{(P/N)}$ of type ($P/N$) for
$\|N \omega - P \|_{d+e}$ small enough.\hfil \breakline
Then, for $k \in \natural, k < \frac{r-1}{\tau}$
we can find $D_k > 0$, such that $2d$ of
the eigenvalues $\lambda_1, \dots, \lambda_{2d}$
of the derivative $Df^N (x_{(P/N)})$ satisfy
$$|\lambda_i -1 | \le D_k \| N{\omega} - P\|_d^{k/2} N ,\ i = 1,\dots ,2d$$
$$(|\lambda_i -1 | \le \tilde D_1 N
\exp ( -\tilde D_2 \| N\omega - P\|_d^{\frac{-1}{2(1+\tau)}}),\ i = 1,\dots, 2d )$$
The remaining $e$ eigenvalues are identically $1$.
\REMARK In \clm(volpres) we consider
periodic orbits not of $f$ itself but a closeby
$f^*$ that is a periodic skew-product over $\torus^e$
with rational rigid rotation $\omega_2$.
The neighborhood of the invariant surface can be thought of
as a neighborhood of the invariant surface in action variables of
an extension of the original map
$\tilde f:\torus^{d+e}\times \real^{d+e} \to \torus^{d+e}\times \real^{d+e}$
such that
$\tilde f|_{\torus^{d+e}\times \real^d} = f$ and
$\tilde f|_{\real^e} = {\rm Id}$. A particular example of such an extension
for the map
$$f(\phi_1, \phi_2, A_1) = (\phi_1 + A_1 + g(A_1, \phi_1, \phi_2),
\phi_2 + \omega_2,
A_1 + g(A_1, \phi_1, \phi_2))$$
is given by
$$\tilde f(\phi_1, \phi_2, A_1, A_2) = (\phi_1 + A_1 + g(A_1, \phi_1, \phi_2),
\phi_2 + A_2,
A_1 + g(A_1, \phi_1, \phi_2),
A_2)$$
at the value of $A_2 = \omega_2$.
Our results cover the case that $f$ admits an invariant surface on
which motion is conjugate to rotation. In \cite{FL92} it was shown that if
$f$ admits an invariant set on which motion is semi-conjugate to rotation
then there are periodic orbits approaching the invariant set under certain
conditions on the Lyapunov exponents of $f$ on the invariant set. We include
the statements of the theorems in \cite{FL92} for completeness.
\CLAIM{Theorem}(residue_hyper1)
(Theorem 2.3 in \cite{FL92}) \hfil \break
Assume $\Gamma$ is a hyperbolic set of rotation vector $\omega$ and that
$\{x_n\}$ is a sequence of periodic points of type $(M_n/N_n)$ such that
${\rm orbit}(x_n)$ converges to $\Gamma$. Then, for sufficiently large $n$,
$|R(x_n)|^{1/N_n} > \lambda > 1$. Actually, if the hyperbolic set has (maximum)
Lyapunov exponent $\gamma$, then $\lim_n R(x_n)^{1/N_n} = e^\gamma$.
\CLAIM{Theorem}(residue_hyper2)
(Theorem 4.3 in \cite{FL92}) \hfil \break
Let $f: M\to M$ be a $C^2$ diffeomorphism leaving invariant the ergodic measure
$\mu$. Assume that, with respect to this measure, $f$ has no zero Lyapunov
exponents. Then, for almost every point $x_0$ in the support of $\mu$,
it is possible to find a sequence $\{ x_n\}_{n=0}^\infty$ of periodic
points that converge to $x_0$. Moreover, the sequence of orbits can be chosen
in such a way that the Lyapunov exponents of $x_n$ converge to the
Lyapunov exponents of $x_0$.
\vskip 15cm
\SECTION Proof of the results
\SUBSECTION The $C^r$ case for symplectic maps
In this section we will consider the case of symplectic maps $f$,
satisfying conditions $(i), (ii)$.
The proof consists of three parts. In the first part we will
construct a normal form in the neighborhood of the invariant surface and
approximate the map in that neighborhood with an integrable mapping.
The distance between our map and the integrable map
can be made $O(\|H\|_d^k)$ where $H$ are the actions in an appropriate
coordinate system, for $k$ depending on the smoothness of the
invariant surface and the type of the rotation vector.
In the second part we will show that in a small enough neighborhood of
the invariant surface the rotation vector of periodic orbits that stay
in the neighborhood cannot differ from the rotation vector of the
invariant surface more than the size of the neighborhood.
%In cases of maps that arise from generating functions
%it was shown (see \cite{LW93} and \cite{BK87})
%that if the integrable map has an invariant surface with a
%rational rotation vector, then some periodic orbits survive
%for small enough perturbation. Moreover, the distance between
%the periodic orbits of the perturbed and the integrable maps is
%bounded by the size of the perturbation. Since in the
%integrable case the distance between the invariant surface and
%a nearby periodic orbit is proportional to the difference of
%their rotation vectors we can find a constant $C_k$ such that
%in a neighborhood of the invariant surface
%$$ \| f - I_k \|_r \le C_k \| N\omega - P\|_d^k
%\EQ(estimate)$$
%where $I_k$ is the integrable map that
%approximates $f$ (where $\|\ \|_r$ is a norm in a space of $C^r$ functions).
The last part is a perturbation argument, that
allows us to estimate the eigenvalues of the
derivative along periodic orbits that stay close to the invariant
surface.
We begin the proof by making a change of variables
to a new system of coordinates,
more convenient for studying a neighborhood of the invariant
surface.
\CLAIM{Proposition}(coord)
Let $f$ as above, $\Gamma$ a $C^r$ invariant surface (which is
a graph of a $C^r$ function $\gamma : \torus^d \to \real^d $)
and $f|_\Gamma$ $C^r$ conjugate to rigid rotation with
rotation vector $\bf \omega$.
Then we can find a symplectic, $C^{r-1}$ mapping $h$ defined in
a neighborhood of $\Gamma$, with a $C^{r-1}$ inverse in a
neighborhood of $\Gamma$ such that
$$
h \circ f \circ h^{-1} (\phi, A) =
(\phi + \omega + Av(\phi, A), A + A^2 u(\phi, A))
\EQ(firstred)
$$
where $A^2$ implies all quadratic combinations of the various $A$'s.
\PROOF
The proof consists of two steps. We first shift the action coordinates
so that $(\phi, 0)$ becomes the invariant surface. Then we use the
conjugacy to rigid rotation to deduce \equ(firstred).
Define $h_1 : \torus^d \times \real^d \hookleftarrow$
by
$$ h_1 (\phi, A) = (\phi , A + \gamma (\phi))$$
Then $h_1$ is in $C^r$, symplectic and sends
$\torus^d \times \{0\}^d$ to the
graph of $\gamma$. Thus $h_1 \circ f \circ h_1^{-1}$
leaves the surface
$\torus^d \times \{0\}^d$ invariant.
$$
h_1 \circ f \circ h_1^{-1} (\phi, A) = (v_1(\phi, A), Au_1(\phi, A))
$$
Since the motion on the surface is $C^r$ conjugate to rigid rotation,
there is a $C^r$ function
$\delta : \torus^d \to \torus^d$ with a $C^r$ inverse
(hence $[D\delta]^{-1}$ exists) such that
$ v_1(\delta(\phi), 0) = \delta (\phi + \omega)$.
We introduce (for $r> 1$) the $C^{r-1}$ symplectic transformation
$$h_2(\phi, A) = (\delta(\phi) , [D\delta]^{-1} A)$$
with
$$h_2^{-1} \circ h_1 \circ f \circ h_1^{-1} \circ h_2 (\phi, A) =
(\phi + \omega + Av_2(\phi, A), Au_2(\phi, A) ).\EQ(transf)
$$
We have
$$\frac{\partial A_i'}{\partial A_j} \bigr |_{A = 0} =
\frac{\partial \phi_i'}{\partial \phi_j} \bigr |_{A = 0} = 0, \quad
i \ne j$$
$$
\frac{\partial \phi_i'}{\partial \phi_j} \bigr |_{A = 0} = 1, \quad \forall
\ i$$
$$
\frac{\partial A_i'}{\partial \phi_j } \bigr |_{A = 0} = 0, \quad \forall \
i, j$$
so, since the map is symplectic
$$\frac{\partial A_i'}{\partial A_i} \bigr |_{A = 0} = 1$$
Moreover, from condition $(ii)$
$ \partial \phi_i ' / \partial A_i \ne 0 $ or $v_2 (\phi, 0) \ne 0$.
This concludes the proof of \clm(coord).
\QED
\REMARK
In the case $d=1$, Birkhoff's theorem guarantees that an invariant curve
of a non-singular symplectic map, with irrational rotation number is
a graph. Birkhoff's theorem fails in the case that the twist condition
(condition $(ii)$ for $d=1$) is violated.
Also for higher
dimensions we are not aware of an analog of Birkhoff's theorem
(in the case $d=2$ there is an analog of Birkhoff's theorem
for a class of maps that can be expressed as a finite number of
compositions of one-dimensional twist maps -- see \cite{Ma91}).
In the general case,
the condition that the invariant curve is a graph over $\torus^d$
can be substituted by a more local condition (weaker in the case of
the maps we have been studying and also applying for singular symplectic
maps -- i.e. maps with zero torsion).
%coordinates can be substituted by a much weaker one.
If $\Gamma$ is homotopic to $\torus^d$ there are coordinates,
in a neighborhood of $\Gamma$ for which the invariant surface reduces to a
graph. Then, condition
$(ii)$ needs only be satisfied in a neighborhood of the
invariant surface, in the transformed coordinates $\equ(transf)$ (i.e.
$v_2(\phi, 0) \ne 0$) for the conclusions of \clm(residue_smooth) to be valid.
%\REMARK
%In the two-dimensional case of twist maps on the annulus, the condition
%that $\Gamma$ is a graph follows from Birkhoff's theorem.
%We are not aware of a similar result for
%the case of higher dimensions. However, one only needs that the invariant
%curve is a graph over (some) angle coordinates.
%Due to the special form of the system we are studying,
%a generalization of Birkhoff's theorem, due to Mather (see \cite{Ma91})
%implies that $\Gamma$ is the graph of a Lipschitz function.
%, (i.e. when motion in the second angle coordinate differs
%from rigid rotation).
We introduce some further notation. In the following we use
$\{ i_m \}$ as a multi-index. The statement $\{ i_m \} $ will denote all
possible combinations of indices $i_{1,m}, \dots, i_{j,m}$ such that
$\sum_l i_{l,m} = m$. Moreover, the expression $A^{\{i_m\}}$ will mean
all possible combinations of the different $A$'s raised to all possible
indices allowed from the condition $\sum_l i_{l,m} = m$. Also, a
symbol $Q_{\{i_m\}}$ ``multiplying'' $A^{\{i_m\}}$ will denote
a multitude of functions, one for each combination of the $A$'s allowed
(e.g. $Q_{\{i_1\}}$ corresponds to $d$ functions,
$Q_{\{i_2\}}$ corresponds to $d(d-1)/2$ functions, etc.)
We can now construct a normal form for $f$ in a neighborhood of the invariant
surface.
We first construct $d$ approximate integrals in a small neighborhood of the
invariant surface.
\CLAIM{Lemma}(normalform)
Let $f\in C^r$ as above and $\bf \omega$ a rotation vector of type ($K, \tau$).
Then, given any $k \in \natural$,
$k < \frac{r-1}{\tau}$, we can find functions
$H_{\{i_0\}}, \allowbreak H_{\{i_1\}},\allowbreak \dots, \allowbreak H_{\{i_k\}} : \torus^d \to \real^d$
and constants $C_k$
such that $H : \torus^d \times \real^d \to \real^d$
$$ H = \sum_{m=0}^k A^{\{i_m\}} H_{\{i_m\}}(\phi)$$
satisfies \hfil \break
\centerline{$ \| H\circ f - H \| \le C_{k+1} \|A\|^{k+1}_d$}
\PROOF
Expanding in $A$ we have
$$\eqalign{
& H \circ f(\phi, A) = \sum_m (A + A^{\{i_2\}} u(\phi, A) )^{\{i_m\}}
H_{\{i_m\}}(\phi + \omega + A v(\phi, A)) \cr
& = \sum_m (A + A^{\{i_2\}} u(\phi, A ) )^{\{i_m\}} \left [
\sum_{l=0}^k c_{\{j_l\}}
\frac{\partial H_{\{i_m\}}}{\partial A^{\{j_l\}}}
(\phi + \omega + A v(\phi, A))|_{A=0} +
O(A^{\{m+k+1\}}) \right ] \cr
& = \sum_{m=0}^k A^{\{i_m\}} [ H_{\{i_m\}}(\phi + \omega)
+ H_{\{i_{m-1}\}} (\phi + \omega) u(\phi, 0 )
+ L_{\{i_m\}} (\phi) ] + O(A^{\{k+1\}})
\cr
}$$
where $c_{\{j_l\}}$ the coefficients of the
Taylor expansion
and $L_{\{i_m\}}$ depends on
$H_{\{i_0\}}, H_{\{i_1\}}, \allowbreak \dots, H_{\{i_{m-2}\}}$
and their
derivatives up to order $m$, as well as on the derivatives of $H_{\{i_{m-1}\}}$.
Notice that changes in $H_{\{i_{m-1}\}}$ by a constant do not affect
$L_{\{i_m\}}$.
Matching terms by order we have
$$\eqalign{
H_{\{i_0\}}(\phi) = & H_{\{i_0\}}(\phi + \omega) \cr
H_{\{i_m\}}(\phi) = & H_{\{i_m\}}(\phi + \omega)
+ H_{\{i_{m-1}\}} (\phi + \omega) u(\phi, 0 )
+ L_{\{i_m\}} (\phi) \cr}
\EQ(homology)
$$
Equations \equ(homology) are of the form
$$g(\phi + \omega) - g(\phi) = f(\phi) \EQ(homol)$$
It is well known (see \cite{SM71}, \cite{Ar88}) that for the case of
$\omega$ of type $(K, \tau)$, given $f \in C^q$ with zero average over
the $d$-torus, there exists $g \in C^{q-\tau}$ that satisfies \equ(homol)
(for $q>\tau$).
%Equations \equ(homology) can be solved successively.
For ${\{i_0\}}$, the only possible
continuous solution is $H_{\{i_0\}} = {\rm constant}$
(from the condition $\int_{\torus^d} L_{\{i_1\}} d\phi = 0 $, if
$\int_{\torus^d} u(\phi, 0) d\phi \ne 0$ we get
$H_{\{i_0\}} = 0$).
If, for $\{i_m\}, m>0$,
$H_{\{i_0\}}, H_{\{i_1\}}, \dots, H_{\{i_{m-2}\}}$ are uniquely determined
and $H_{\{i_{m-1}\}}$ is determined up to a constant, then $L_{\{i_m\}}$
is completely
determined. Moreover, $H_{\{i_m\}}$ can be determined up to a constant if
and only if
$$\int_{\torus^d} [L_{\{i_m\}}(\phi) + u(\phi, 0)
H_{\{i_{m-1}\}} (\phi + \omega) ] d\phi = 0
\EQ(condition)$$
which uniquely determines the average value of $H_{\{i_{m-1}\}}$ in
the case $\int_{\torus^d} u(\phi, 0) \ne 0$. In the case
$\int_{\torus^d} u(\phi, 0) = 0$ we can show that the choice
$\int_{\torus^d} H_{\{i_m\}}(\phi) = 0,\ m\ge 0$ satisfies \equ(condition).
Consider the truncation $H^{[\le m-1]} =
\sum_{m=0}^{m-1} A^{\{i_m\}} H_{\{i_m\}}(\phi)$, satisfying
\equ(homology) up to order $m-1$. Then we have
$$\int_{\torus^d}\{ H^{[\le m-1]} (\phi, A) - H^{[\le m-1]}\circ f(\phi, A)\}
d\phi = 0$$
since $f$ symplectic implies $f$ preserves volume in phaser-space.
We have
$$\eqalign{H^{[\le m-1]} (\phi, A) - H^{[\le m-1]}\circ f(\phi, A) = &
A^{\{i_m\}} \left( L_{\{i_m\}}(\phi) +
u(\phi, 0) H_{\{i_{m-1}\}} (\phi + \omega) \right) \cr
& + O(A^{\{i_{m+1}\}})}$$
thus, condition \equ(condition) is satisfied.
%(since, from the form of $f$, $\int_{\torus^d} u(\phi, 0) d\phi \ne 0$).
%(since, $f$ symplectic implies $f$ preserves volume in phase-space and after
%a change of variables, \equ(condition) can be shown to be satisfied).
The process can,
inductively, be carried out as long as $L_{\{i_k\}}$ is smooth
enough (at least $C^{\tau +\epsilon}$). Since in every
step of the induction the
smoothness of $L_{\{i_k\}}$ decreases by $\tau$, we have the bound
$k \tau > r - 1$ or $ k < \frac{r-1}{\tau}$. If $f$ is $C^\infty$
or analytic the induction can be carried out for all $k\in \natural$.
\QED
We have constructed $d$ functions $H$ that are approximate integrals in the
vicinity of the invariant surface. Since $H_{\{i_0\}} = 0$, $H$ is
a small perturbation of $A$ and the surface $H=h$, for $\|h\|_d$
small, is topologically nontrivial.
Defining
$$ \bar H (h) = \int_{H=h} A d\phi $$
the function $\bar H$ is conserved under $f$ up to $O(\|A\|_d^{k+1})$
in a neighborhood of $A=0$.
We change coordinates, in such a way that $\bar H$ replaces $A$,
using a generating function $S$
$$S(\Phi, A) = ( A + \int_{\torus^d} \sum_{m=2}^k A^{\{i_m\}}
H_{\{i_m\}} (s) ds)\Phi \EQ(generating)$$
The function $S$ generates the symplectic transformation
$$\eqalign{
\bar H & = D_1S(\Phi, A) =
A + \int_{\torus^d} \sum_{m=2}^k A^{\{i_m\}} H_{\{i_m\}}(s) ds \cr
\phi & = D_2S(\Phi, A) = \Phi(1 + \frac{\partial}{\partial A}\int_{\torus^d}
\sum_{m=2}^k A^{\{i_m\}} H_{\{i_m\}}(s) ds)\cr
}\EQ(transformation)$$
In the new coordinates %the map $f$ takes the form
$$
f(\Phi, \bar H) = (\Phi + \omega + \bar H \Delta(\bar H), \bar H)
+ E(\Phi, \bar H)\EQ(almost_int)$$
where the remainder satisfies (in appropriate norms)
$\|E\| \le C_k \|\bar H\|_d^{k+1}$ and
$\Delta(0) \ne 0$.
\QED
\REMARK Another way to construct the normal form would be to perform
successive canonical transformations (for example using the method
of Lie transforms) and reduce $f$ to an integrable map, up to
$O(A^{\{k+1\}})$, in a neighborhood of the invariant surface.
The method of successive canonical
transformations has been used in the case $d=1$ in
\cite{McK92}, whereas the method of constructing an approximate integral
in \cite{FL92}. We favor the method of
constructing approximate integrals, since it lends itself
to efficient numerical implementations.
In the case that the map $f$ is analytic, our estimates hold in a complex
neighborhood of $\torus^d \times \{ 0\}^d$ of the form
$\{ |\Im \Phi_i | < \xi, |\bar H_i| \le \xi, \quad i = 1,\dots,d\}$
for some $\xi > 0$.
In the new $(\Phi, \bar H)$ coordinates, we have
$\|DE\| \le C_k\|\bar H\|_d^k$ and
$$ Df (\Phi, \bar H) =
\pmatrix{1&F(\bar H)\cr
0&1\cr
}
+ O(\|\bar H\|_d^k)
\EQ(derivative)
$$
where $F(\bar H) = \Delta(\bar H) + \bar H\Delta' (\bar H)$.
In a neighborhood of the invariant surface only periodic orbits with rotation
vectors close to the rotation vector of the invariant surface are allowed.
Since $F(0) \ne 0$, using the implicit function theorem, we conclude
that the actions $\bar H_{\rm per}$
of a periodic orbit of period $N$ in the vicinity of
the invariant surface are bounded by
$$ C_1 \| N\omega - P\|_d \le \|\bar H_{\rm per}\|_d \le C_2 \| N\omega -P\|_d$$
The existence of periodic orbits for maps that are close to integrable
(such as map \equ(almost_int) in a neighborhood of the invariant surface)
has been studied in the case where $f$ has a generating function, in
\cite{BK87} and \cite{LW93}. It was shown that some periodic orbits
of the integrable system persist, for small enough perturbation, and
their distance from the original periodic orbits can be bounded by the
size of the perturbation. Although in \cite{LW93} only Hamiltonian
flows were considered (which correspond to maps with a generating function)
the methods used could be easily extended to periodic orbits of symplectic
maps that do not have a generating function.
The last part of the proof consists of a simple perturbative argument.
Since we are interested in
the eigenvalues of the derivative along
periodic orbits, we estimate the norm of products
of matrices close to the ones appearing in \equ(derivative).
\CLAIM{Lemma}(matrices)
Let $\{ A_i\}_{i=1}^N $ be a set of $2 d\times 2 d$ matrices of the form
$$A_i = \pmatrix{1&a_i\cr
0&1\cr
}
$$
with
$$\max \left(1, \sup_{1\le i\le N} (\sup_{1\le l,k \le d}|(a_i)_{lk}|) \right )
\le A$$
and $\{ B_i\}_{i=1}^N$
satisfy
$$\sup_{\scriptstyle 1\le i\le N\atop\scriptstyle 1\le j,k\le 2d} |(B_i)_{jk} -
|(A_i)_{jk}|\le\epsilon \quad \hbox{\rm with }\epsilon < A\ .$$
Then, all the eigenvalues of $B= B_1 \ldots B_N$,
$\lambda_1, \dots, \lambda_{2d}$ satisfy
$$ |1-\lambda_i| \le 2\{(1+3d\sqrt{A}\sqrt{\epsilon})^N -1\}$$
\PROOF
We introduce the following norms for vectors and matrices: for a vector in
$\real^{2d}$
we define $\| v\|_\delta = \sum_{i=1}^d (|v_i|\delta + |v_{i+d}|)$
and for any $2d\times 2d$ matrix $C$,
$\|C\|_\delta = \sup_{v\in \real^{2d}} \|Cv\|_\delta / \|v\|_\delta$.
Then, if $\lambda$ is an eigenvalue of $C$, we have
$|\lambda| \le \| C\|_\delta$.
%Moreover,
%$$\eqalign{
%\left\|\pmatrix{C_{11}&C_{12}&C_{13}&C_{14}\cr
% C_{21}&C_{22}&C_{23}&C_{24}\cr
% C_{31}&C_{32}&C_{33}&C_{34}\cr
% C_{41}&C_{42}&C_{43}&C_{44}\cr
% }\right\|\le
%\max ( & |C_{11}|+ |C_{21}|+ |C_{31}|\delta + |C_{41}|\delta, \cr
% & |C_{12}|+ |C_{22}|+ |C_{32}|\delta + |C_{42}|\delta,\cr
% & |C_{13}|\delta^{-1}+ |C_{23}|\delta^{-1}+ |C_{33}|+ |C_{43}|,\cr
% & |C_{14}|\delta^{-1}+|C_{24}|\delta^{-1} +|C_{34}|+ |C_{44}| ) \cr
%}
%$$
For the matrices $A_i, B_i$ and for $\delta < 1$
$$\eqalign{
\|A_i\|_\delta &\le 1 + d \max(1, |(a_i)_{jk}|) \delta \le 1 + d A\delta \cr
\|A_i - B_i\|_\delta & \le \epsilon\max (d + d\delta, d+ d\delta^{-1}) =
\epsilon d (1 + \delta^{-1})\cr
}
\EQ(matestim)
$$
To prove the claim about the eigenvalues of $B$, notice that the eigenvalues of
$B-I$, $\mu_1, \dots, \mu_{2d}$ satisfy
$$|\mu_i| \le \|B - A_1 \cdots A_N + A_1 \cdots A_N - I\|_\delta
\le \| B - A_1 \cdots A_N\|_\delta + \|A_1 \cdots A_N - I\|_\delta$$
We write
$$B= B_1 \ldots B_N = (A_1 + (B_1 -A_1))(A_2 + (B_2 -A_2))\dots (A_N + (B_N-A_N))$$
Expanding, grouping terms, %and using the estimates \equ(matestim)
we get
$$\eqalign{
B = A_1\cdots A_N & + \sum_i A_1\cdots A_{i-1} (B_i -A_i)A_{i+1}\cdots
A_N \cr
& + \sum_{i,j} A_1 \cdots A_{i-1} (B_i - A_i) A_{i+1} \cdots A_{j+1}
(B_j -A_j) A_{j+1} \cdots A_N \cr
& + \cdots\cr
& + (B_1-A_1)\cdots(B_N - A_N)\cr}
$$
or
$$\eqalign{\|B - A_1 \cdots A_N \|_\delta \le
& {N\choose 1} \max_i \|A_i\|_\delta^{N-1} \|B_i-A_i\|_\delta \cr
&+ {N\choose 2} \max_i \| A_i \|_\delta^{N-2} \|B_i -A_i\|_\delta^2 \cr
& + \cdots \cr
& + {N\choose N} \max_i \|B_i -A_i\|_\delta^N\cr
}
$$
and using the estimates \equ(matestim)
$$\|B - A_1\dots A_N\|_\delta \le
[ 1 + dA\delta + d(1 + \delta^{-1})\epsilon ]^N -
[1+d \delta A]^N $$
Choosing $\delta = (\epsilon/A)^{1/2} < 1$ we obtain
$$\|B - A_1\dots A_N\|_\delta \le
[1 + 3d \sqrt{A} \sqrt{\epsilon} ]^N -1
\EQ(resbound)$$
Similarly $\|A_1\cdots A_N - I\|_\delta$ can be bounded, following the
same steps as above by
$$\|A_1\cdots A_N - I\|_\delta \le (1 + d\sqrt{A} \sqrt{\epsilon} )^N - 1$$
Since $\mu_i = \lambda_i -1 $ we have
$$|\lambda_i - 1| \le
(1 + 3d \sqrt{A} \sqrt{\epsilon} )^N +
(1 + d \sqrt{A} \sqrt{\epsilon} )^N -2
\le
2\{(1 + 3 d \sqrt{A} \sqrt{\epsilon} )^N -1\} $$
\QED
Putting all the estimates together, for $N$ large enough, we
can bound all the eigenvalues of $Df^N({\bf x})$
for a $(P/N)$ periodic orbit by
$$|\lambda_i - 1| \le D_k \|N{\bf \omega} - P\|_d^{k/2} N $$
This concludes the proof of
\clm(residue_smooth).
\QED
\SUBSECTION The analytic case
To prove \clm(residue_anal) we only need to compute the values of the constants
$C_k, D_k$ and choose the best value for $k$. %in the bound \equ(resbound) we
%need to extract the $k$ dependence from $C_k$ and find the $k$
%value for which the bound becomes optimal.
The optimal bound depends on the diophantine properties of the rotation
vector $\bf \omega$.
In this section we use the following norms for analytic
functions over a complex neighborhood
${\cal T}_\delta =
\{ (\phi, A) |
\Re \phi_i \in [0,1], |\Im \phi_i | \le \delta, |A_i| \le \delta\}$
of the invariant surface
$$ \| F \|_\delta \equiv \sup_{{\cal T}_\delta} |F|$$
or, if $F$ denotes several functions
$$ \| F \|_\delta \equiv \max_i \| F_i \|_\delta$$
We first state a lemma that provides quantitative bounds for the solution
to equations similar to \equ(homol).
\CLAIM{Lemma}(homol-lem)
Let $L$ be a bounded analytic function on ${\cal T}_\delta$ and assume
$L$ has zero average over $\torus^d$. For $\omega$ diophantine of
type $(K, \tau)$ we can find a solution of the equation \hfil \break
\centerline{$H(\phi) - H(\phi +\omega) = L(\phi)$}
unique, up to an additive constant, on ${\cal T}_\delta$.
Moreover, the solution is bounded on any smaller domain
${\cal T}_{\delta -\eta}$ by \hfill \break
\centerline{$\| H \|_{\delta -\eta} \le C_{K, \tau, d} \eta^{-\tau} \|L\|_\delta$}
for any $0 < \eta < \delta$.
A proof of \clm(homol-lem) can be found in \cite{R\"us76}, \cite{Ar88},
\cite{FB89}.
In the process of constructing $d$ approximate integrals in the
neighborhood of the invariant surface we need to solve the
equations
$$H_{\{i_m\}} (\phi) - H_{\{i_m\}} (\phi + \omega) =
H_{\{i_{m-1}\}} (\phi +\omega) u(\phi, 0) + L_{\{i_m\}}(\phi) $$
where
$L_{\{i_m\}} (\phi ) = L_{\{i_m\}}^1(\phi) - L_{\{i_m\}}^2(\phi)$
with
$$\eqalign{
L_{\{i_m\}}^1(\phi) = & \sum_{j = 1}^m \frac{1}{\{i_j\}!} \left(
\frac{\partial}{\partial A}\right) ^{\{i_j\}} H_{\{i_{m-j}\}} (\phi +
\omega + Av(\phi,A)) |_{A=0} \cr
L_{\{i_m\}}^2(\phi) = & \sum_{j = 2}^m H_{\{i_{m-j}\}}(\phi)
\frac{1}{\{i_j\}!} \left(\frac{\partial}{\partial A}\right) ^{\{i_j\}}(A
+ A^{\{i_2\}} u (\phi, A))^{\{i_j\}} |_{A=0}\cr
}$$
under the condition \equ(condition).
We will use induction to estimate bounds on the $H$'s.
\CLAIM{Theorem}(recursion-bounds)
If the invariant surface is analytic in ${\cal T}_\delta$ and $\omega$ is
diophantine of type $(K, \tau)$ then
$$\eqalign{
\|\tilde H_{\{i_m\}} \|_{\delta -m\eta} & \le ED^m \cr
\max | \bar H_{\{i_m\}} | & \le ED^m \cr
}$$
where $\bar H = \int_{\torus^d} H d\phi$, $\tilde H = H -\bar H$,
$\delta - k \eta > 0$ and $D = \tilde K \eta^{-1-\tau}$ for
$\tilde K, E$ numbers that depend on the system, the invariant surface,
the dimension and $\omega$.
\PROOF
Using induction, the hypothesis holds for $m=1$. Assuming
that all $H_{\{i_m\}}$'s are determined completely up to order $m-2$
and up to an additive constant for $H_{\{i_{m-1}\}}$ and satisfy the
bounds in the assumption we have
$$
\sup_{\|A\|_d \le \eta/2V} \| H_{\{i_{m-j}\}} (\phi + \omega +
Av(\phi, A)\|_{\delta-(m-1/2)\eta} \le
\| H_{\{i_{m-j}\}}\|_{\delta -(m-1)\eta} \le
\| H_{\{i_{m-j}\}}\|_{\delta -j\eta}
$$
where $V = \sup_{{\cal T}_\delta} |v(\phi, A)|$.
Using Cauchy estimates to bound derivatives with respect to $A$
(see \cite{PW94} for a justification of Cauchy estimates for the
case
of max norms in $\complex^d$) we have
$$
\eqalign{
\sup_{\|A\|_d \le \eta/2V} \| \frac{1}{\{i_j\}!} \left(
\frac{\partial}{\partial A} \right)^{\{i_j\}} H_{\{i_{m-j}\}} (\phi +\omega
+Av(\phi,A)) |_{A=0} \|_{\delta - (m-1/2)\eta}\cr
\hfil \le \|H_{\{i_{m-j}\}}\|_{\delta -j\eta} \frac{(2V)^j}{\eta^j}
}$$
and
$$
\sup_{\|A\|_d \le \eta/2V} \| \frac{1}{\{i_j\}!} \left(
\frac{\partial}{\partial A} \right)^{\{i_j\}} (A+A^{\{i_2\}}u(\phi,
A))^{\{i_j\}} |_{A=0} \|_{\delta - (m-1/2)\eta}
\le \frac{1}{\eta^j}$$
>From the above estimates we deduce
$$\eqalign{
\|L_{\{i_m\}}^1\|_{\delta -(m-1/2)\eta} &\le D^{m-1} E \frac{4V}{\eta} \cr
\|L_{\{i_m\}}^2\|_{\delta -(m-1/2)\eta} &\le D^{m-1} E \frac{2}{\eta} \cr
}$$
>From the condition \equ(condition)
$$\|\bar H_{\{i_{m-1}\}} \| \le ED^{m-1}$$
for $\eta$ fixed and $E$ large enough.% ($E \prop \eta^{-1}$.
Using \clm(homol-lem) and fixing $\eta \le \delta /2k$ we have
$$\|\tilde H_{i_m}\|_{\delta-m\eta} \le ED^{m-1} \tilde K \eta^{-1-\tau}\le ED^m
$$
which concludes the induction.
\QED
To conclude the proof of \clm(residue_anal) we fix $\eta = \delta/2k$
and have $C_k \le \tilde K \left(\frac{k}{\delta}\right)^{k(1+\tau)}$
and, using a simple maximization argument over $k$,
$$\max_{k \in \natural} \left(\frac{k}{\delta}\right)^{k(1+\tau)} B^k
\le \exp [-(1+\tau)B^{-1/(1+\tau)} \delta e^{-1} ]$$
Letting $B = \|N\omega - P\|_d^{1/2}$ concludes the proof of
\clm(residue_anal).
\QED
\REMARK
\clm(residue_anal) is also valid for the case of complex maps with complex
invariant surfaces, as long as the non-degeneracy condition $(ii)$ is
satisfied in a neighborhood of the invariant surface.
\SUBSECTION The volume-preserving case
The proof for the case of a quasi-periodic perturbation
of a symplectic map is similar to the proofs of \clm(residue_smooth)
and \clm(residue_anal). We sketch the proof (referring to the proofs in
sections 3.1 and 3.2) and
emphasize the differences.
We study invariant sets of maps
$f : \torus^{d+e}\times \real^d \to \torus^{d+e}\times \real^d$
on which motion is conjugate to rigid rotation with
rotation vector $\omega = (\omega_1, \omega_2)$, ($\omega_1 \in \torus^d$,
$\omega_2 \in \torus^e$),
with
$$f(\phi_1, \phi_2, A) = (f_1(\phi_1, \phi_2, A), \phi_2+\omega_2)$$
where $f_1 : \torus^{d+e}\times \real^d \to \torus^d\times \real^d$
and $f_1(\cdot, \phi_2, \cdot)$ is symplectic.
The first part of the proof consists of constructing a normal form
for $f$ in a neighborhood of the invariant surface with
rotation vector $\omega$. As in \clm(coord)
we can
find a map $h$, defined in a neighborhood of the invariant surface,
such that
$$ h \circ f \circ h^{-1} (\phi_1, \phi_2, A) =
(\phi_1 + \omega_1 + A_1 v(\phi_1, \phi_2, A_1), \phi_2 + \omega_2,
A_1 + A_1^2 u(\phi_1, \phi_2))
$$
with $v(\phi_1, \phi_2, 0) \ne 0$.
We can now construct $d$ approximate integrals for $f$
in a neighborhood of the invariant surface, by expanding
and matching by orders as in \clm(normalform).
The difference at this point is that not only the
properties of $\omega_1$ (the rotation vector for the
symplectic coordinates) but also the combined properties of
$\omega_1$ and $\omega_2$ are important.
After constructing the approximate integrals,
we perform a transformation (using a generating function in the
``symplectic'' coordinates, identity in the
remaining coordinates) to substitute the approximate integrals
for the original ``actions''.
%Another important
%difference with the symplectic case is that, although $f^*$
%has $d+e$ ``actions'' we can construct only $d$ approximately
%conserved integrals. Fortunately, due to the special nature
%of $f^*$ the ``actions'' $A_2$ are exactly conserved.
The normal form for $f$ in a neighborhood of the invariant
surface is
$$
f
(\Phi_1, \phi_2, \tilde A_1) =
(\Phi_1 + \omega_1 + \tilde A_1 \Delta(\tilde A_1), \phi_2 + \omega_2,
\tilde A_1) +
(E_1(\Phi_1, \phi_2, \tilde A_1), 0_{e}, E_2(\Phi_1, \phi_2, \tilde A_1 ))$$
where $\Delta(0, \omega_2) \ne 0$ and
$\|E_{1,2}\| \le C_k \| \tilde A_1 \|_d^{k+1}$
in appropriate norms.
Instead of studying the normal form for
$f$ itself we will study the extension
$f^* : \torus^{d+e}\times \real^{d+e}\to \torus^{d+e}\times \real^{d+e}$
with
$$
\eqalign{f^*(\Phi_1, \phi_2, \tilde A_1, A_2) = &
(\Phi_1 + \omega_1 + \tilde A_1 \Delta(\tilde A_1), \phi_2 + A_2,
\tilde A_1, A_2) \cr
& +
(E_1(\Phi_1, \phi_2, \tilde A_1), 0_{e},
E_2(\Phi_1, \phi_2, \tilde A_1), 0_{e})
\cr}
$$
The map $f^*$ is also area preserving and, for $A_2 \equiv \omega_2$,
motion in the $\Phi_1, \phi_2, \tilde A_1$ coordinates
under $f^*$ is identical to motion in the $\Phi_1, \phi_2, \tilde A_1$
coordinates under $f$. The map
$f^*$ has the advantage that in a neighborhood of an invariant surface
with rotation vector of type $(K, \tau)$ one can find
periodic orbits (by simply changing $A_2$ to nearby rational numbers).
%On the other hand, there are no periodic orbits for $f$ with
%$\omega_2$ irrational.
The bounds on the eigenvalues of the derivative follow
from \clm(matrices). The $2e$ eigenvalues corresponding
to rotation in the $\phi_2, A_2$
coordinates are identically $1$.
Following arguments similar to section 3.2
we can also reproduce the proof for the analytic case.
This concludes the proof of \clm(volpres).
\QED
\REMARK In the case of a general volume-preserving map
$f:\torus^d \times \real \to \torus^d \times \real$ under conditions
similar to the ones in \clm(volpres) it is possible to
construct one approximate integral in the neighborhood of the
invariant surface. However no
result similar to \clm(volpres) is possible, since we have no control
for the motion along the angle coordinates, similar to what we have for
the symplectic case.
\SECTION Conclusions
Our results in \clm(residue_smooth), \clm(residue_anal) and
\clm(volpres) suggest that the
eigenvalues of the derivative of a symplectic map along a periodic orbit,
are in higher dimensions an analog of the residue (as used in Greene's
criterion for two-dimensional twist maps). Based on this analogy,
an efficient numerical algorithm can be implemented to indicate existence
of a close-by invariant surface. Since convergence to the limit behavior
(either 1 for the case of an invariant surface or $\infty$ for the case
of a uniformly hyperbolic invariant set) is exponentially fast, relatively
low period orbits can be used. In a separate paper we implement such an
algorithm
for the case of a quasi-periodic excitation of a two-dimensional
symplectic map (see \cite{T95}).
%for the case of a four-dimensional symplectic map (see \cite{T95}).
%symplectic map (see \cite{T95}).
Periodic orbits can also be used to investigate behavior at breakdown.
If transition can be described in terms of a fixed point of a
renormalization group operator with a co-dimension one stable manifold,
the eigenvalues of the periodic orbits scale with the period of the
orbit and the distance from breakdown. Recently, Kosygin constructed
a renormalization group operator and showed that if, under repeated action
of the operator, the map converges to a -- trivial -- fixed point, then
the original map admits an invariant surface (see \cite{Kos91}). No such
description is known for the behavior at breakdown. Numerical studies
and analytical arguments suggest that if such a renormalization
operator exists, there are regions in parameter space where
behavior at breakdown is governed by dynamics more
complex than a simple fixed point (see \cite{MMS94, ACS91, T95}).
Another interesting problem is to determine the existence of
lower-dimensional hyperbolic tori on which motion is conjugate to
rigid rotation with a resonant rotation vector.
One can separate phase space in the neighborhood of the low-dimensional
torus to the center manifold of the torus and the hyperbolic directions.
Arguments similar to the ones we used in this paper can be used
to show that along the center manifold the map is close to an integrable
normal form. Along the hyperbolic directions behavior can be described
using arguments similar to \cite{FL92}.
%One could reduce the
%motion in a neighborhood of the invariant torus to a normal form where
%the map will be approximated in some directions by a map
%with the same hyperbolic behavior and in the directions along the
%torus (and their canonical conjugates) by a normal form similar -- but
%lower dimensional -- to the one we introduced in the proof of
%\clm(residue_smooth).
The natural result appears to be that $2d^*$
eigenvalues (where $d^*$ the dimension of the low-dimensional torus)
of the derivative of the map along periodic orbits will approach
1, while the rest will approach $e^{\lambda_i T}$ where $\lambda_i$
the non-zero Lyapunov exponents of the orbits on the
low-dimensional torus and $T$ the period.
Unfortunately a numerical algorithm
to estimate domains of existence of lower dimensional hyperbolic tori,
would be difficult to implement,
since we can not numerically isolate the eigenvalues that
tend to 1, from eigenvalues that become exponentially large.
\SECTION Acknowledgments
Most of this work was completed at the University of Toronto.
I would like to thank Rafael de la Llave, Claudio Albanese, Luis Seco,
Jeff Xia, Jim Meiss and Robert MacKay
for useful discussions and suggestions during this work.
\SECTION References
%\ref\no{Au83}\by{S. Aubry}\paper{The twist map, the extended Frenkel-Kontorova model and the devil's staircase}\jour{Physica D}\vol{7}\pages{240--258}\yr{1983}\endref
\ref\no{ACS91}\by{R. Artuso, G. Casati, D. L. Shepelyansky}\paper{Breakdown of universality in renormalization dynamics for critical invariant torus}\jour{Europhys. Lett.}\vol{15}\pages{381--386}\yr{1991}\endref
%\ref\no{Ar63}\by{V.I. Arnol'd}\paper{Proof of a theorem of A. N. Kolmogorov on the invariance of quasi periodic motions under small perturbations of the Hamiltonian}\jour{Russ. Math. Surveys}\vol{18}\pages{9}\yr{1963}\endref
\ref\no{Ar88}\by{V.I. Arnol'd}\book{Geometrical Methods in the Theory of Ordinary Differential Equations, 2$^{\rm nd}$ ed.}\publisher{Springer-Verlag}\yr{1988}\endref
\ref\no{BK87}\by{D. Bernstein, A. Katok}\paper{Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians}\jour{Invent. Math.}\vol{88}\pages{225--241}\yr{1987}\endref
%\ref\no{BPV90}\by{N. Buric, I.C. Percival, F. Vivaldi}\paper{Critical function and modular smo\-oth\-ing}\jour{Nonlinearity}\vol{3}\pages{21--37}\yr{1990}\endref
%\ref\no{CC88}\by{A. Celletti, L. Chierchia}\paper{Construction of analytic K.A.M. surfaces and effective stability bounds}\jour{Comm. Math. Phys.}\vol{118}\pages{119--161}\yr{1988}\endref
\ref\no{FB89}\by{F. Fasso, G. Benettin}\paper{Composition of Lie transforms with rigorous estimates and applications to Hamiltonian perturbation theory}\jour{J. Appl. Math. Phys. (ZAMP)} \vol{40}\pages{307--329}\yr{1989}\endref
\ref\no{FL92}\by{C. Falcolini, R. de la Llave}\paper{A rigorous partial justification of Greene's criterion}\jour{Jour. Stat. Phys.} \vol{67}\pages{609--643}\yr{1992}\endref
%\ref\no{FL92-1}\by{C. Falcolini, R. de la Llave}\paper{A rigorous partial justification of Greene's criterion}\jour{Jour. Stat. Phys.} \vol{67}\pages{609--643}\yr{1992}\endref
%\ref\no{FL92-2}\by{C. Falcolini, R. de la Llave}\paper{Numerical Calculation of domains of analyticity for perturbation theories in the presence of small divisors}\jour{Jour. Stat. Phys.} \vol{67}\pages{645--666}\yr{1992}\endref
%\ref\no{Gol94}\by{C. Gol\'e}\paper{Optical Hamiltonians and symplectic twist maps}\jour{Physica D}\vol{71}\pages{185--195}\yr{1994}\endref
\ref\no{Gr79}\by{J.~M.~Greene}\paper{A method for de\-ter\-mi\-ning a sto\-cha\-stic tran\-si\-ti\-on}\jour{J. Math. Phys.}\vol{20}\pages{1183--1201}\yr{1979}\endref
%\ref\no{KO86}\by{S. Kim, S. Ostlund}\paper{Simultaneous rational approximations in the study of dynamical systems}\jour{Phys. Rev.}\vol{34A}\pages{3426}\yr{1986}\endref
%\ref\no{Kol54}\by{A.N. Kolmogorov}\paper{On the conservation of conditionally periodic motions under small perturbations of the Hamiltonian}\jour{Dokl. Akad. Nauk.}\vol{SSR 98}\pages{469}\yr{1954}\endref
\ref\no{Kos91}\by{D.V. Kosygin}\paper{Multidimensional KAM theory for the renormalization group viewpoint}\inbook{Dynamical Systems and Statistical Mechanics}\bybook{Ya. G. Sinai}\jour{Adv. Sov. Math.}\pages{99-129}\publisher{American Mathematical Society}\yr{1991}\endref
%\ref\no{Ll93}\by{R. de la Llave}\paper{Introduction to KAM theory}\jour{Archived in \break {$\tt mp\_arc@math.utexas.edu$}, \#93-8}\yr{1993}\endref
%\ref\no{LR91}\by{R. de la Llave, D. Rana}\paper{Accurate strategies for K.~A.~M. bounds and their implementation}\inbook{Computer Aided proofs in Analysis}\bybook{K. Meyer, D. Schmidt}\publisher{Springer Verlag}\yr{1991}\endref
\ref\no{LW93}\by{R. de la Llave, C.E. Wayne}\paper{Whiskered and low dimensional tori in nearly integrable Hamiltonian systems}\jour{Preprint}\yr{1993}\endref
\ref\no{Ma91}\by{J. N. Mather}\paper{Variational construction of orbits of twist diffeomorphisms}\jour{Jour. Amer. Math. Soc.}\vol{4}\pages{207--263}\yr{1991}\endref
\ref\no{McK82}\by{R.S. MacKay}\paper{Renormalization in area preserving maps}\jour{Princeton thesis}\yr{1982}\endref
\ref\no{McK92}\by{R.S. MacKay}\paper{On Greene's residue criterion}\jour{Nonlinearity}\vol{5}\pages{161--187}\yr{1992}\endref
\ref\no{MMS94}\by{R.S. MacKay, J.D. Meiss, J. Stark}\paper{An approximate renormalization for the break-up of invariant tori with three frequencies}\jour{Phys. Lett. A}\vol{190}\pages{417--424}\yr{1994}\endref
%\ref\no{Mo62}\by{J. Moser}\paper{On invariant curves of area preserving maps of an annulus}\jour{Nach. Akad. Wiss., Gottingen, Math. Phys. Kl. II}\vol{1}\pages{1}\yr{1962}\endref
%\ref\no{Mu89}\by{M. Muldoon}\paper{Ghosts of order on the frontier of chaos}\jour{Caltech thesis}\yr{1989}\endref
\ref\no{Po93}\by{H. Poincar\'e}\book{New methods of celestial mechanics, D. Goroff (ed.)}\publisher{AIP}\yr{1993}\endref
\ref\no{PW94}\by{A.D. Perry, S. Wiggins}\paper{KAM tori are very sticky: rigorous lower bounds on the time to move away from an invariant Lagrangian torus with linear flow}\jour{Physica D}\vol{71}\pages{102--121}\yr{1994}\endref
%\ref\no{Ran87}\by{D. Rana}\paper{Proof of accurate upper and lower bounds to stability domains in small denominator problems}\jour{Princeton thesis}\yr{1987}\endref
\ref\no{R\"us76}\by{H. R\"ussmann}\paper{Note on sums containing small divisors}\jour{Commun. Pure Appl. Math}\vol{29}\pages{755}\yr{1976}\endref
\ref\no{SM71}\by{ C. L. Siegel, J. Moser}\book{Lec\-tu\-res on Ce\-les\-tial Me\-cha\-nics}\publisher{ Sprin\-ger-Ver\-lag, New York}\yr{1971}\endref
\ref\no{T95}\by{S. Tompaidis}\paper{Numerical study of invariant sets of a volume-preserving map}\jour{Preprint}\yr{1995}\endref
\end