\documentclass[12pt]{article}
\topmargin -2cm
\oddsidemargin=-0.7cm
\rightmargin=3cm
\textwidth=16cm
\textheight=24cm
\title{A remark on ``Some remarks on the problem of ergodicity of the
Standard Map'' }
\author{V.F. Lazutkin\footnote{St.-Petersburg State University,
Ulyanov.str.,1, Petrodvorets,
St.-Petersburg, 198904 (RUSSIA)}}
\date{\relax}
\newtheorem{theorem}{Theorem}
\newtheorem{claim}{Claim}
\newtheorem{statement}{Statement}
\newtheorem{remark}{Remark}
\newtheorem{conjecture}{Conjecture}
\begin{document}
\maketitle
\centerline{AMS classification scheme numbers:
34C35, 58C05, 58C15, 70K50}
\vspace{1.5cm}
\noindent
{\bf Abstract} We consider the standard family
$F_g:(x,y)\mapsto (g\cos(2\pi x)-y,-x)$ of area-preserving maps
defined on the two-torus ${\bf R}^2/{\bf Z}^2\,,$
the parameter $g$ ranging along the real axis ${\bf R}\,.$
We prove that there exists a subset ${\cal E}\subset{\bf R}$ whose density
tends to zero along the real axis, such that for any $g$ in the complement to
${\cal E}$ the map $F_g$ is ergodic an has nonzero
Lyapunov exponents almost everywhere
in ${\bf R}^2/{\bf Z}^2$ with respect to the Haar measure.
\bigskip
\bigskip
This note is an immediate continuation of the paper \cite{gl}.
We investigate the map $F_g:{\bf R}^2/{\bf Z}^2\to{\bf R}^2/{\bf Z}^2$
given by equations $F_{g}(x,y)=(x_1,y_1)$ where
\begin{eqnarray}
x_1&=&g\cos(2\pi x)-y\ {\rm mod}\, 1\,,\label{map}\\
y_1&=&x\ {\rm mod}\, 1\,,\nonumber
\end{eqnarray}
Our goal is to prove the following
\begin{theorem}
There exists a subset ${\cal E}\subset {\bf R}$ and a constant $C>0$
such that
\begin{itemize}
\item[(i)] the following estimate is true:
\begin{equation}
leb({\cal E}\cap [p/2,(p+1)/2])\le C\,|p|^{-1}\,,
\end{equation}
for all $p\in{\bf Z}$ (here $leb$ denotes the Lebesgue measure on ${\bf R}$);
\item[(ii)] if $g\in{\bf R}\setminus{\cal E}\,,$
then the map $F_g$
is ergodic, and
its Lyapunov exponents are nonzero almost everywhere in
in ${\bf R}^2/{\bf Z}^2$ with respect to the Haar measure.
\end{itemize}
\label{th}
\end{theorem}
Let us recall in brief the construction, suggested in \cite{gl}.
Evidently, it is sufficient to consider only sufficiently
large positive values of $g\,.$
The phase space can be reduced by means of additional symmetry
generated by the maps
$S_1:(x,y)\mapsto (1/2-x,1/2-y){\rm mod}\, 1\,,\,
S_2:(x,y)\mapsto (-x,y+1/2){\rm mod}\, 1\,,\,
S_3:(x,y)\mapsto (x+1/2,-y){\rm mod}\, 1\,.$ The quotient space is
the projective plane which can be represented by means of of
a fundamental domain which is the square $[-0.25,0.25]\times [-0.25,0.25]$ with
edges glued by the maps
\begin{equation}
g_1:(-0.25,y)\mapsto (0.25,-y)\ \ {\rm and}\ \
g_2:(x,-0.25)\mapsto (-x,0.25)\,.
\label{glueing}
\end{equation}
The angle points $(\pm 0.25,\pm 0.25)$ represent a hyperbolic periodic orbit
of period 2 whose stable and unstable manifolds are the basic elements of
our construction. There exist initial arcs, $W^s_0$ and $W^u_0$ which
connect the points $(-0.25,0.25)$ and $(0.25,-0.25)$ with a point $H$
on the diagonal $x=y$ of the square close to $(0.25,0.25)\,,$
the following equality being fulfilled:
$R(W^s_0)=W^u_0$ where $R$ is a reversor defined by the equations
$(x,y)\mapsto(y,x)\,.$ Denote $W_n^u=F^n(W_0^u)\,, n=1,2\dots,$
and $W_n^s=F^{-n}(W_0^s)\,, n=1,2\dots.$ The partition
${\cal P}=\{A_1,A_2,\dots,A_N$ of the phase space created by
the cells of the web
$W_0^u\cup W_0^s\cup W_1^s$ gives rise to a symbolic dynamics:
almost each point $z$ of the phase space acquires an infinite code
$(\dots,n_{-2}(z),n_{-1}(z),n_{0}(z),n_1(z),n_2(z),\dots)$ by the rule:
$F_g^i(z)\in A_{n_i(z)}\,, i\in {\bf Z}\,.$
In \cite{gl} we constructed a decreasing sequence ${\rm FLE}(n), n=1,2,\dots,$
of subsets of the phase space with the following properties:
\begin{enumerate}
\item ${\rm FLE}(n)$ is a union of some cells of $W_k^s\cup W_k^u$;
\item the complement to $\bigcup_{r=-n+1}^{r=n-1}F_g^r({\rm FLE}(n))$
possesses a cone structure: there are fields of invariant stable and
unstable cones (see \cite{w}).
\end{enumerate}
A crucial role in proving Theorem \ref{th} plays the estimate
\begin{equation}
leb({\rm FLE}(n))\le C_1\mu^{n+1}
\label{est}
\end{equation}
where $C_1>0$ and $0<\mu<1$ do not depend on $n\,.$ Of course (\ref{est})
is not true for all values of the parameter $g\,.$ An obstacle to (\ref{est})
is the existence of so-called ``islands of stability'' created by KAM
circles. Indeed, a periodic chain of islands of positive measure
possesses at least one component inside ${\rm FLE}(n)\,.$ In \cite{gl}
we discovered the following mechanism of creating islands.
The cells of $W_k^s\cup W_k^u$ which belong to ${\rm FLE}(n)$
form a kind of one-dimensional structure placed in a vicinity of
the coordinate axes $x=0$ and $y=0\,.$ The picture is symmetric
with respect to the reversor $R$ (reflection with respect to the diagonal),
so we can restrict ourselves with considering only the horizontal branch
of ${\rm FLE}(n)\,.$ Its cells can be aggregated in ``units'', $c_k\,,
k=(k_1,k_2,\dots,k_n)$ consisting of all points $z\in {\rm FLE}(n)$
such that $n_{i-1}(z)=k_i\,, i=1,2,\dots,n\,.$ We found in \cite{gl}
that each $c_k$ contains a ``central spot'', $w^*_k\subset c_k\,,$
whose sizes are much less than those of $c_k$ and which possess
the following property: if $c_k$ contains an island such that its
$n$-th iterate contains in the vertical part of ${\rm FLE}(n)$
then the mentioned island is contained in $w^*_k\,.$
The sizes of $w^*_k$ admit the estimates:
\begin{equation}
{\rm the \ width\ of}\ w^*_k\sim g^{-1-2n}\,,\ \
{\rm the \ height\ of}\ w^*_k\sim g^{-1-n}\,,
\label{sizes3}
\end{equation}
It is not difficult to prove that $F^n(w^*_k$ is contained in a cell $c'$
of the vertical part of ${\rm FLE}(n)$ such that
$R(c')\subset c_{\overline{k}}$ where $\overline{k}$ is the ``inverse''
sequence to $k$:
$\overline{k}=(\sigma(k_n),\sigma(k_{n-1},\dots\sigma(k_{1})$ where
$\sigma:\{1,2,\dots,N\}\to \{1,2,\dots,N\}$ is the involution defined
by the equalities: $RF_f(A_i)=A_{\sigma(i)}\,, i=1,2,\dots,N\,.$
A necessary condition for the existence of an island of the considered
type reads
\begin{equation}
w^{\prime\prime*}_{\overline{k}}\cap w^*_l\ne\emptyset
\label{condition3}
\end{equation}
where $w^{\prime\prime*}_{\overline{k}}=F^{n+1}_g(w^{*}_{k})\,,$
a small horseshoe, and $ w^*_l$ is another central spot with
the code $l\,.$
In \cite{gl} we tried to destroy {\it all} collisions of the form
(\ref{condition3}) by throwing out intervals on the axis of the
parameter $g$ for which the collisions take place.
Thus we obtained the estimate
\begin{equation}
leb({\cal E}_n)\le const\cdot g^{-1}
\label{badest}
\end{equation}
where ${\cal E}_n)$ is the union of all intervals
in $[p/2,(p+1)/2]$ containing the values of
$g$ for which (\ref{condition3}) occurs in ${\rm FLE}(n)\,.$
This does not suit to our aim because the sum of the right hand sides of
(\ref{badest}) for all $n$ diverges and cannot be small
To overcome this difficulty
we formulated in \cite{gl} a conjecture which concerns possible codes
of cells containing islands. Unfortunately this conjecture hardly can
be true.
In the present note we make another step: we eliminate
the {\it double } collisions of the form
\begin{equation}
w^{\prime\prime*}_{\overline{k}}\cap w^*_l\ne\emptyset
\qquad {\rm and}\qquad RF_g^{-1}(w^{*}_k)\cap w^*_m\ne\emptyset\,.
\label{condition4}
\end{equation}
If $n\ge 2\,,$ any periodic chain of islands of considered type
has a fragment contained in the spots entering (\ref{condition4})
and their iterates. Hence the condition (\ref{condition4}) also is necessary
for existing islands. Denote by ${\cal E}^{(2)}_n$ the set of all $g$
in the interval $[p/2,(p+1)/2]$ such that (\ref{condition4}) fails
for all triples $m,l,m$ of multi-codes of the length $n\,.$
\begin{statement}
The following estimate is true uniformly for all $n$ and $p$:
\begin{equation}
leb({\cal E}^{(2)}_n)\le \left(C_3p\right)^{-n}
\label{goodest}
\end{equation}
where $C_3$ is a positive number.
\label{st}
\end{statement}
Theorem \ref{th} follows from Statement \ref{st} if we set
${\cal E}=[-R,R]\cup {\cal E}'\cup(-{\cal E}')$ where
$R$ is a sufficiently large positive number and
${\cal E}'=
{\cal E}_1\cup\bigcup_{n=2}^{\infty}{\cal E}^{(2)}_n\,.$
If $g$ does not belong to ${\cal E}$ then $F_g$ does not contain islands,
and there is no obstacle for (\ref{est}) to be true. The rigorous proof
of (\ref{est}) is cumbersome. The mentioned cone structure
on the complement to ${\rm FLE}(n)$ and its forward and backward iterates
up to the order $n-1$ gives rise to the positive Lyapunov exponents.
\medskip
\noindent
{\it An idea of the proof of Statement \ref{st}.}
Denote by $x_k$ the $x$-coordinate of the center of $w_k^*\,.$
Let $g=p/2+t$ where the variable $t$ ranges through the interval $[0,1/2]\,.$
In view of (\ref{map}) and (\ref{sizes3})
the condition (\ref{condition4}) is equivalent to the following
Diophantine equations:
\begin{eqnarray}
x_{\overline{k}}+x_l&=&t+O\left(p^{-1-2n}\right)\,,\label{e1}\\
x_k+x_m&=&t+O\left(p^{-1-2n}\right)\,.\label{e2}
\end{eqnarray}
We have to evaluate the measure of the set of the values of the parameter
$t$ for which the equations (\ref{e1}) and (\ref{e2}) admit a solution.
The problem can be interpreted as a dynamical problem in the following
manner: how much time the ``diagonal'' trajectory on ${\bf R}^2$ given by the
equation $t\mapsto (t,t)$ stays in the
$O\left(p^{-1-2n}\right)$-neighbourhood of the discrete set
${\cal D}=\{z_{(k,l,m)}=(x_{\overline{k}}+x_l, x_k+x_m):
(k,l,m)\in [0,N]^{3n}\}\subset
{\bf R}^2\,.$
If we fix the value of $k$ the corresponding subset ${\cal D_k}$
of ${\cal D}$ becomes a two-dimensional discrete nonlinear
``fractal lattice'' whatever such words mean. Locally it can
be approximated by a linear discrete fractal of the form
$$
x=x^0+\sum_{j=1}^n(l_i-l_i^0)\prod_{i=1}^j\lambda_i^{-1}\,,\qquad
y=y^0+\sum_{j=1}^n(m_i-m_i^0)\prod_{i=1}^j\mu_i^{-1}\,,
$$
where $\lambda_i$ and $\mu_i$ are kinds of multiplicators
along the trajectories starting with the centers of $w_l^*$
and $w_m^*$ respectively (the formulae for the multiplicators can
be extracted from the paper \cite{l}, but approximately $\lambda_i=
-2\pi g\sin(2\pi x_i)$ where $x_i$ is the first coordinate of
the $i$-th point of the mentioned trajectory).
The linear problem can be resolved by successive application of
the ergodic theorem to a particle moving in a rectangular lattice,
the edges of which can be chosen incommensurate due to an appropriate
choice of the center of the linearisation. The result should be
the following: the time which the particle stays in the mentioned
neighbourhood of the lattice asymptotically (if $g$ tends to infinity)
is equal to the relative area of the neighbourhood for the majority
of the initial data (=values of $k$). Since the lattice is two-dimensional
and contains $O\left(p^{2n}\right)$ points,
it gives the following estimate: if we denote by ${\cal E}_{n,k}$
the mentioned above time then
$$
leb( {\cal E}_{n,k})=\left[O\left(p^{-1-2n}\right)\right]^{2}
\times O\left(p^{2n}\right)
$$
Multiplying by the number of possible choices of the multi-vector
$k$ that is by the number of the order $p^n\,,$ gives us the desired
estimate (\ref{goodest})
\medskip
The detailed proof is forthcoming.
\bigskip
\noindent
{\bf Acknowledgments}
This investigations was partially supported
by INTAS grant 97-0771 and grant 97-14.3-43 of Higher Education
State Committee of Russian Federation
\vskip 0.5 truecm
\def\refname{References}
\begin{thebibliography}{99}
\bibitem{gl} A. Giorgilli and V.F. Lazutkin, Some remarks on the problem
of ergodicity of the Standard Map, Physics Letters A 272 (2000), pp.359-367.
Electronic version of the paper has been archived in
$mpej.unige.ch/mp_arc/index.html/\#99-287$ or
$mp_arc@fireant.ma.utexas.edu/\#99-287$
\bibitem{l} V.F. Lazutkin,
Positive entropy for the standard map I. Pr\'epublications 94-47
Universit\'e de Paris-Sud Math\'ematiques B\^atiment 425
91405 ORSAY France (1994), 34 pp.
\bibitem{w} M. Wojtkowski, Invariant families of cones and Lyapunov exponents.
Ergodic Theory and Dynamical Systems
vol.5, No.1, 145--161, 1985.
\end{thebibliography}
\end{document}