% This TeX file was generated by ./texpsinc
% from the original TeX file pap.tex
% and the PostScript files pic1.eps.
% TeX writes out the included PostScript to files.
%
% Here is part of pap.tex:
\documentstyle[12pt,epsf]{article}

% Here are the
% TeX macros for dumping included Postscript to files.
% Adapted from Knuth's \answer macro in the TeXbook.
% Jamie Stephens, jamies@math.utexas.edu, 28 Nov 94

\def\endofps{EndOfTheIncludedPostscriptMagicCookie}
\chardef\other=12
\newwrite\psdumphandle 
\outer\def\psdump#1{\par\medbreak
  \immediate\openout\psdumphandle=#1
  \copytoblankline}
\def\copytoblankline{\begingroup\setupcopy\copypsline}
\def\setupcopy{\def\do##1{\catcode`##1=\other}\dospecials
  \catcode`\\=\other \obeylines}
{\obeylines \gdef\copypsline#1
  {\def\next{#1}%
  \ifx\next\endofps\let\next=\endgroup %
  \else\immediate\write\psdumphandle{\next} \let\next=\copypsline\fi\next}}
\outer\def\closepsdump{
  \immediate\closeout\psdumphandle}

% Here is the PostScript for pic1.eps:
\message{Writing file pic1.eps}
\psdump{pic1.eps}%!PS-Adobe-3.0 EPSF-3.0
%%Creator: (ImageMagick)
%%Title: (pic1.eps)
%%CreationDate: (Fri Jul 30 12:05:50 1999)
%%BoundingBox: 7 97 417 507
%%DocumentData: Clean7Bit
%%LanguageLevel: 1
%%Pages: 0
%%EndComments

%%BeginDefaults
%%PageOrientation: Portrait
%%EndDefaults

%%BeginProlog
%
% Display a color image.  The image is displayed in color on
% Postscript viewers or printers that support color, otherwise
% it is displayed as grayscale.
%
/buffer 512 string def
/byte 1 string def
/color_packet 3 string def
/pixels 768 string def

/DirectClassPacket
{
  %
  % Get a DirectClass packet.
  %
  % Parameters:
  %   red.
  %   green.
  %   blue.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile color_packet readhexstring pop pop
  compression 0 gt
  {
    /number_pixels 3 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add 3 mul def
  } ifelse
  0 3 number_pixels 1 sub
  {
    pixels exch color_packet putinterval
  } for
  pixels 0 number_pixels getinterval
} bind def

/DirectClassImage
{
  %
  % Display a DirectClass image.
  %
  systemdict /colorimage known
  {
    columns rows 8
    [
      columns 0 0
      rows neg 0 rows
    ]
    { DirectClassPacket } false 3 colorimage
  }
  {
    %
    % No colorimage operator;  convert to grayscale.
    %
    columns rows 8
    [
      columns 0 0
      rows neg 0 rows
    ]
    { GrayDirectClassPacket } image
  } ifelse
} bind def

/GrayDirectClassPacket
{
  %
  % Get a DirectClass packet;  convert to grayscale.
  %
  % Parameters:
  %   red
  %   green
  %   blue
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile color_packet readhexstring pop pop
  color_packet 0 get 0.299 mul
  color_packet 1 get 0.587 mul add
  color_packet 2 get 0.114 mul add
  cvi
  /gray_packet exch def
  compression 0 gt
  {
    /number_pixels 1 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add def
  } ifelse
  0 1 number_pixels 1 sub
  {
    pixels exch gray_packet put
  } for
  pixels 0 number_pixels getinterval
} bind def

/GrayPseudoClassPacket
{
  %
  % Get a PseudoClass packet;  convert to grayscale.
  %
  % Parameters:
  %   index: index into the colormap.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile byte readhexstring pop 0 get
  /offset exch 3 mul def
  /color_packet colormap offset 3 getinterval def
  color_packet 0 get 0.299 mul
  color_packet 1 get 0.587 mul add
  color_packet 2 get 0.114 mul add
  cvi
  /gray_packet exch def
  compression 0 gt
  {
    /number_pixels 1 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add def
  } ifelse
  0 1 number_pixels 1 sub
  {
    pixels exch gray_packet put
  } for
  pixels 0 number_pixels getinterval
} bind def

/PseudoClassPacket
{
  %
  % Get a PseudoClass packet.
  %
  % Parameters:
  %   index: index into the colormap.
  %   length: number of pixels minus one of this color (optional).
  %
  currentfile byte readhexstring pop 0 get
  /offset exch 3 mul def
  /color_packet colormap offset 3 getinterval def
  compression 0 gt
  {
    /number_pixels 3 def
  }
  {
    currentfile byte readhexstring pop 0 get
    /number_pixels exch 1 add 3 mul def
  } ifelse
  0 3 number_pixels 1 sub
  {
    pixels exch color_packet putinterval
  } for
  pixels 0 number_pixels getinterval
} bind def

/PseudoClassImage
{
  %
  % Display a PseudoClass image.
  %
  % Parameters:
  %   class: 0-PseudoClass or 1-Grayscale.
  %
  currentfile buffer readline pop
  token pop /class exch def pop
  class 0 gt
  {
    currentfile buffer readline pop
    token pop /depth exch def pop
    /grays columns 8 add depth sub depth mul 8 idiv string def
    columns rows depth
    [
      columns 0 0
      rows neg 0 rows
    ]
    { currentfile grays readhexstring pop } image
  }
  {
    %
    % Parameters:
    %   colors: number of colors in the colormap.
    %   colormap: red, green, blue color packets.
    %
    currentfile buffer readline pop
    token pop /colors exch def pop
    /colors colors 3 mul def
    /colormap colors string def
    currentfile colormap readhexstring pop pop
    systemdict /colorimage known
    {
      columns rows 8
      [
        columns 0 0
        rows neg 0 rows
      ]
      { PseudoClassPacket } false 3 colorimage
    }
    {
      %
      % No colorimage operator;  convert to grayscale.
      %
      columns rows 8
      [
        columns 0 0
        rows neg 0 rows
      ]
      { GrayPseudoClassPacket } image
    } ifelse
  } ifelse
} bind def

/DisplayImage
{
  %
  % Display a DirectClass or PseudoClass image.
  %
  % Parameters:
  %   x & y translation.
  %   x & y scale.
  %   label pointsize.
  %   image label.
  %   image columns & rows.
  %   class: 0-DirectClass or 1-PseudoClass.
  %   compression: 0-RunlengthEncodedCompression or 1-NoCompression.
  %   hex color packets.
  %
  gsave
  currentfile buffer readline pop
  token pop /x exch def
  token pop /y exch def pop
  x y translate
  currentfile buffer readline pop
  token pop /x exch def
  token pop /y exch def pop
  currentfile buffer readline pop
  token pop /pointsize exch def pop
  /Helvetica findfont pointsize scalefont setfont
  x y scale
  currentfile buffer readline pop
  token pop /columns exch def
  token pop /rows exch def pop
  currentfile buffer readline pop
  token pop /class exch def pop
  currentfile buffer readline pop
  token pop /compression exch def pop
  class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
  grestore
} bind def
%%EndProlog
%%Page:  1 1
%%PageBoundingBox: 7 97 417 507
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\closepsdump

% Finally, here is the rest of pap.tex:
% \documentstyle[12pt,epsf]{article}
\topmargin -2cm
\oddsidemargin=-0.7cm
\rightmargin=3cm
\textwidth=16cm
\textheight=24cm
\title{ Some remarks on the problem of ergodicity of the Standard Map}
\author{A. Giorgilli\footnote{Dipartimento di matematica e
Applicazioni, Universit\`a di Milano Bicocca, Via Bicocca degli
Arcimboldi 8, 20126\ \ MILANO (Italy)}
\ and
V.F. Lazutkin\footnote{St.-Petersburg State University,
Ulyanov.str.,1,  Petrodvorets,
St.-Petersburg,  198904 (RUSSIA)}}

\date{\relax}
\newtheorem{claim}{Claim}
\newtheorem{statement}{Statement}
\newtheorem{remark}{Remark}
\newtheorem{conjecture}{Conjecture}
%\def\qed=\Box
\begin{document}
\maketitle
%\centerline{(University of Milano Bicocca and St.-Petersburg State University)}

 
%\hspace{1.5cm}
% Ulyanov.str.,1,  Petrodvorets, St.-Petersburg}, 198904, {\it Russia.}

%\hspace{1.5cm}E-mail address}:lazutkin@mph.phys.spbu.ru

\vspace{1.5cm}

\centerline{AMS classification scheme numbers: 
34C35, 58C05, 58C15, 70K50}

\vspace{1.5cm}

\noindent
{\bf Abstract} We consider the problem of removing the islands
of stability in the phase space of the standard map by means of
tuning the parameter. A possible construction which gives
a classification of periodic  chains of islands in terms of a symbolic dynamics
and predicts the values of the parameters for which the island
with a given symbolic code exists is suggested and discussed. 


\vspace{1.5cm}
\section{Introduction}


Recently O.Knill issued a preprint \cite{K} where he proves that the
metric entropy of the Standard map of 2-dimensional torus,
${\bf T}^2={\bf R}^2/{\bf Z}^2\,,$
given (in our notations) by the equations  $F_{g,M}(x,y)=(x_1,y_1)$ where
\begin{eqnarray}
x_1&=&Mx+g\cos(2\pi x)-y\ {\rm mod}\, 1\,,\label{map}\\
y_1&=&x\ {\rm mod}\, 1\,,\nonumber
\end{eqnarray}
is positive for any integer $M$ and for all large enough values of the
parameter $g\,.$ Nevertheless the problem of finding the values of
the parameter $g$ for which the map (\ref{map}) is ergodic (and
isomorphic to a Bernoulli shift) remains still open. It seems probable
that the method, used by Knill, which do not involve the study of the
dynamics of the map itself, cannot be applied to solve this problem.

In this paper we try to formulate another approach, based on the study
of the partition of the phase space created by drawing suitable arcs
of the stable and unstable manifolds of some periodic points (see
\cite{Laz3}). 
  
Before getting into the details, let us dwell on the basic notions.
Numerical experiments show that there is a nonvoid open subset
${\cal E}={\cal E}(M)\,,$ of the real line, such that $F=F_{g,M}$ with
$g\in {\cal E}$ possesses an {\it island} of stability. This means
that there is an invariant subset of positive Lebesgue measure
filled up with periodic chains of KAM circles
(see \cite{arn}, \cite{Koz}, \cite{Laz1}, \cite{Mos}) . Recall that a
{\it periodic chain of KAM circles } is an invariant set, $C\,,$ 
of the form $C=\bigcup_{k=0}^{p-1}F^k(C_0)\,,$ where $C_0$ is 
a smoothly embedded one-dimensional torus, 
${\bf T}^1={\bf R}/{\bf Z}\,,$ and $F^p\vert_{C_0}$ is smoothly
conjugated to a shift: $\xi\mapsto\xi+\omega\, {\rm mod} 1\,,$
with an irrational ({\it e.g.} Diophantine) rotational number $\omega\,.$ 
The number $p$ is the {\it period} of $C\,,$ the circles
$C_k=F^k(C_0)\,, k=0,1,\dots,p-1.$ are the {\it KAM circles,} or
the {\it components} of $C\,.$ 


Of course, if the map possesses
an island it cannot be ergodic. How large is ${\cal E}\,?$ It has
been proven \cite{D} that it is dense in $[a,+\infty]$ for
sufficiently large $a\,.$ So the problem of ergodicity reduces
essentially to the estimation of the ${\it leb}({\cal E}\cap [n,n+1])$
for large values of $n\in {\bf N}\,.$ Here  ${\it leb}$ stands for the
Lebesgue measure on the real line. 

%To solve it we have to focus on
%the question of existing of islands in the phase space of
%the map (\ref{map})for large values of $g\,.$    

In this paper we suggest a method which provides us with a
classification of islands and, in some na\"\i ve way,
gives a necessary condition of existing of island of a given
type. An essential ingredient of the construction is the notion
of a {\it Fundamental Locus of Ellipticity (FLE).} Let $F$ be 
a diffeomorphism of a surface $Z$ onto itself.
 Suppose that $F$ preserves
a measure $\mu$ with positive smooth density with respect to
the Lebesgue 2-dimensional measure in each local chart.
Then all definitions and assertions of KAM theory are true
for such $F\,.$ 
\smallskip

{\sl
A subset $L\subset Z$ is called FLE if it contains at 
least one component of each periodic chain of KAM circles.
}
\smallskip

We are going now to formulate a sufficient condition for a subset 
to be a FLE.
Let $Z$ be equipped with a Riemannian metric. Let $\alpha\in]0,2\pi [\,.$
A {\it cone}  with an {\it opening} $\alpha$ at a point $z\in Z$ 
is a subset
of the tangent space to $Z$ at $z$ of the form:
$\{\vec v\ne 0\,: \vert\angle(\vec v,\vec v_0)\vert<\alpha/2\}\,,$
where $\vec v_0$ is a fixed nonzero vector.
The image of a cone at $z$ with respect to the tangent map to $F$
is again a cone at $F(z)$ but possibly with another opening.
 Let $\Lambda\subset Z$ be a (not necessarily
invariant) subset.  
A family of cones, $K_z\subset T_zZ\,, z\in\Lambda\,,$  is called
a {\it uniformly shrinking cone field}
 on $\Lambda$ with respect to the map $F$ if
there exists a number $0<q <1$ such that the following is true:
\smallskip

{\sl for any point $z\in\Lambda\,,$ such that $F(z)\in\Lambda\,,$
the image of $K_z$ with respect to the tangent map to $F$ is contained
strictly in $K_{F(z)}\,,$ and the ratio of the openings of the image
of $K_z$ 
and $K_{F(z)}$ is less than $q\,.$}
\smallskip

We say that $\Lambda$ is {\it uniformly hyperbolic} if it possesses
two families of cones: $K_z^u\,, z\in\Lambda\,,$ the unstable one,
which is  uniformly shrinking with respect to $F\,,$ and
$K_z^s\,, z\in\Lambda\,,$ the stable one,
which is  uniformly shrinking with respect to $F^{-1}\,.$  

One can easily deduce the
following sufficient condition for a subset $L$ to be a FLE:  

\begin{statement} Let $L\subset S$ possesses the properties:
\begin{enumerate}
\item if a KAM circle has a nonvoid intersection with $L\,,$
it is  wholly contained in $L\,;$
\item there is a natural $r$ such that the complement to 
$\bigcup _{k=-r}^{r}F^k(L)$ is uniformly hyperbolic.
\end{enumerate} 
Then $L$ is a FLE. 
\end{statement}

{\it Proof.} If a periodic chain of KAM circles lies completely in
$\Lambda\,,$ it possesses a uniform hyperbolic structure
(see \cite{CW}, \cite{W}). This contradicts to the quasiperiodicity
of the dynamics induced by the iterates $F$ on it. QED
\medskip

Note that, if the boundary of an open set $L$ consists of arcs
of the stable and unstable manifolds of some periodic hyperbolic
points of $F\,,$ then $L$ satisfies automatically to the condition
1 of the Statement 1.

Further in this paper we try to construct FLE and derive consequences
concerning the existence of island of different types.

Suppose we succeeded in constructing  a decreasing sequence,
$L_1\supset L_2\supset\dots\,,$ of FLE such that $\mu(L_n)\to 0$
as $n\to\infty\,.$ Then, as it follows from the preceding discussion,
 the map $F$ does not possess islands at all.

\section{A model map}

We will illustrate the notions defined in Introduction taking the map
\ref{map} as a model. To make it simpler we put $M=0\,.$ The map
$F_{g,0}$
possesses the following periodic trajectory of period 4
(we use the representation of the torus by means of the square
$[-0.5,0.5]\times[-0.5,0.5]$ with glued opposite sides) :
\begin{equation}
(-1/4,1/4)\mapsto (-1/4,-1/4)\mapsto (1/4,-1/4)\mapsto (1/4,1/4)\,.
\label{pertraj}
\end{equation}
This trajectory is hyperbolic provided $g>0\,.$ In our further
constructions we will use the arcs of the stable and unstable 
manifolds of (\ref{pertraj}).

Since we are going to make in some sense a global analysis
of the situation with islands of stability, it seems reasonable
to reduce the phase space by means of a symmetry. Consider the group
$G$ of transformations of ${\bf T}^2$ generated by the maps
$S_1:(x,y)\mapsto (1/2-x,1/2-y){\rm mod}\, 1\,,\, 
S_2:(x,y)\mapsto (-x,y+1/2){\rm mod}\, 1\,,\,
S_3:(x,y)\mapsto (x+1/2,-y){\rm mod}\, 1\,.$
It is a symmetry group of $F_{g,0}$ in the sense that $F_{g,0}$ maps
orbits of $G$ to orbits of $G$
(see(\cite{Laz4}) for the details). Note that each point of ${\bf T}^2\,,$
different from those belonging to (\ref{pertraj}), lies on an
orbit of $G$ which consists of four points while the set
(\ref{pertraj})
consists of two two-points-orbits  : $\{(-1/4,1/4),(1/4,-1/4)\}$ and 
 $\{(-1/4,-1/4),(1/4,1/4)\}\,.$

Denote by $Z$ the space of the orbits of $G\,.$ It can be obtained
by gluing together the points of the torus belonging to the same
orbit. The space $Z$ can be represented  by means of the 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 space $Z$ is not a smooth surface
because of two singular orbits described above: they produce
${\bf Z}_2$ points $A=\{(-1/4,1/4),(1/4,-1/4)\}$
and $B=\{(-1/4,-1/4),(1/4,1/4)\}$
of the orbifold $Z\,.$  As a topological space, $Z$ is
homeomorphic to the projective plane.

Consider the map $F:Z\to Z$ induced by $F_{g,0}$ on $Z\,.$ It
will play a role of our model in the rest of this paper.
The map $F$ has a hyperbolic (if $g>0$) periodic orbit $\{A,B\}$
of period 2. We will use the stable and unstable manifolds
(separatrices) of $\{A,B\}$ as a building material for
constructing FLE.

The map $F$ has a reversor $R:(x,y)\mapsto(y,x)\,,$ which satisfies
the relations: $R^2=Id$ and $RFR=F^{-1}\,.$ 
  
One has to notice that all the constructions and conclusions can
be extended to  more general families of maps, including 
those consisting of finite differentiable maps.


\section{FLE(1)}

Denote by $W^u_0$ the arc of the unstable manifold of the point $A$
which goes from $A$ up to
the first intersection, $H\,,$ with the diagonal $x=y$ (see Fig.1).
The symmetric arc, $W^s_0=R(W^u_0)\,,$ is an arc of the stable
manifold of the point $A\,.$ These two arcs are seen on Fig.1 as lines
which are running correspondingly close to the upper and right
 edges of the square $[-0.25,0.25]\times[-0.25,0.25]\,,$ and meet 
at the point $H\,,$ the latter being situated close to the right
upper corner. We will use the following notations for the
iterates of these beginning arcs: $W_k^u=F^k(W_0^u)\,, k=1,2\dots,$
and $W_k^s=F^{-k}(W_0^s)\,, k=1,2\dots.$



\begin{figure}[h]
\epsfysize=12.0cm
\centerline{
\leavevmode\epsffile{pic1.eps} 
}     
\caption{The picture of FLE(1) in the window $[-0.25,0.25]\times
[-0.25,0.25]$
for $g=12.12\,.$}
\end{figure}  

 Fig.1 represents the web 
$W_0^u\cup W_1^u\cup W_0^s\cup (W_1^s)$ which creates
a partition of $Z$ into {\it cells.} Some cells of this partition are
filled by grey. They constitute FLE(1). The reason why we selected
these cells is following. Let $c$ be a white cell, that is, not
belonging to FLE(1). Then $F(c)$ is a straight thin ''parallelogram''
 which goes around $Z$ horizontally
and intersects all lines of $F_{-1}(W_0^s)$ {\it transversally,}
 its vertical sides are mapped by $F$ 
into $W^s_0\,,$ while $F(c)$ approaches  $W^u_0$ from
{\it  both} of its sides.     
The analogous assertion is true with respect to $F^{-1}(c)$ and
$W^u_0\,,$ the former being thin  ''parallelograms''  going around $Z$
horizontally. The behaviour of grey cells with respect to the 
application of $F$ and $F^{-1}$ is quite different. Let $c$ be a 
 typical cell which belongs to the vertical string of the grey cells.
Then $F(c)$ is a ''horseshoe'', the images of the vertical edges
of $c$ still belong to  $W^s_0\,,$ while $F(c)$ approaches  $W^u_0$
{\it only from one} side. Analogous assertion is true with
respect to the backward images of the typical cells belonging to
the horizontal string. There are four exceptional cells which are not
''typical'': their positions on Fig.1 are marked by the letter E.
The exceptional cells have either two or six edges. One can glue each
pair of adjacent exceptional cells to obtain a normal cell,
let us call it  a {\it section,} which has two vertical edges made 
of $W_1^s$ and two horizontal edges made of $W_1^u\,.$ Then we
can operate with this section in the same manner as with typical cells. 



\begin{statement} The union of white cells in the preceding
description is hyperbolic.
\end{statement}
\medskip

\noindent
{\it Proof.} All white cells, with one exception, are small
quadrangles
which are close to  parallelograms, while the restrictions of the 
maps $F$ and $F^{-1}$ can be approximated by linear maps.
The only exception is the cell in the right upper corner of
Fig.1, but in this case we can prolong $W_0^u$ and $W_0^s$ a little
bit up to the first intersection with $F^{-1}(W_0^s)$ and 
$F(W_0^u)$ correspondingly, and divide this ''bad'' cell into
three ''good'' ones. Let $c$ be a ''good'' white cell.
Draw its two diagonals. The point $D$ of the intersection of the
diagonals belongs, evidently, to $c$ and the vectors 
tangent to the diagonals at point $D$ create two complementary
cones in the tangents space at $D\,.$ By means of the parallel
translation we spread these pair of cones onto the whole cell
$c\,.$ The horizontal cones constitute the field of the unstable
cones, the vertical cones are the stable ones. 
The description of white cells given above results
in deduction that these cones fields are stretching ones
with respect to $F$ and $F^{-1}$ correspondingly, the stretching
constants being estimated from above by ${\rm const}/\sqrt{g}\,.$
To make this proof effective we have added to FLE(1)two additional
pairs of ''geometrically good'' cells in the vicinity of the
points close to homoclinic tangency (marked by letter E on Fig.1). 
QED
\medskip

One can give an equivalent local description of cells which
belong to FLE(1). Note first that all cells which
possess at least one edge contained in $W_0^u\cup W_0^s$ is
''good'', that is it has no intersection with FLE(1).
 Consider cells with all edges made from
$W_1^u\cup W_1^s\,.$ Let us supply the curves $W_1^u$ and 
$W_1^s$ with the orientation in the sense that a local
observer living at a point belonging to a curve knows which
side of surrounding surface is ``right'' and which one is
''left''. Note that, due to the character of the gluing
functions (\ref{glueing}) the sides change their sense
each time when the curve crosses the border of the
fundamental square $[-0.25,0.25]\times [-0.25,0.25]$ and
appears at the opposite side of it. Then the '' geometric''
test for a cell $c$ to be ''good'' is that $c$ must
have exactly four edges: two made from $W_1^u$ and another two 
from $W_1^s\,,$ and the following orientation rule must be fulfilled:
the cell $c$ acquires the opposite orientation 
senses from the edges of $W_1^u$
and the same rule is for the edges of $W_1^s\,.$ This means, for
example, that $c$ is at the ''right hand side'' with respect to
one of two unstable  edges and at the ''left hand side''
with respect to another. All god cells satisfy this geometric test.
 
There is another description of cells belonging to FLE(1).
Consider the horizontal vector field, $\frac{\partial}{\partial x}\,,$
on the torus. It cannot be projected onto $Z$ as a vector field,
but it can be done with the corresponding field of projective
directions.
It is not difficult to deform slightly the horizontal field so
that $W_0^u$ becomes an integral line of the projected field
of projected directions, denote the latter by $X^u\,,$ and its image
with respect to the reversor $R$ by $X^s\,.$ Consider the function
$\Delta_1$
whose value at a point $z$ is the square of the sinus of  the angle
between $F_*(X^u)$ and $F^{-1}_*(X^s)$ both evaluated at $z\,.$
The gray cells are those which contain zeroes of $\Delta_1$ 
(cf.\cite{GLS} where the corresponding zeroes constituted the lines
of folds ). More precisely, the line of zeroes of $\Delta_1$
goes close to the axes $x=0$ and $y=0\,.$ It passes trough each grey cell
and has a quasi-intersection at  the central cell.

\section{Islands inside FLE(1)}

Here we consider the islands whose iterates never quit FLE(1).
The {\it central island} is  the collection of all KAM circles which
are contained, together with all their iterates, in the central cell
which is painted by dark grey in Fig.1. Of course, the KAM circles of
the central island cannot intersect with the boundary of the cell.
Neither they do intersect with additional lines if we draw some
$W^u_k$ and $W^s_k\,.$ For small values of $k\,,$ say $k=2,3$ 
the intersection of these additional lines with the central cell look
like straight lines or parabolas of the form $x=C_1-2\pi^2gy^2$
and $y=C_2-2\pi^2gx^2$ (the coefficient at the square comes from
the equation \ref{map} where  $cos$ is replaced with its
Taylor quadratic approximation). Consider a more general situation
where we have two families of parabolas of the form
 \begin{equation}
  x=\frac12K_1y^2+C_1\,,\qquad y=\frac12K_2x^2+C_1\,,
  \end{equation}
  where $C_1$ and $C_2$ are the parameters of the families, and
  $K_1$ and $K_2$ are the curvatures which are supposed to be 
  large constants. One can make a change of variables  of the form
  \begin{equation}
     x=A\xi\,,\qquad y=B\eta
  \end{equation}
  where $A$ and $B$ are constants so that in new coordinates
  the parabolas of both families have maximal curvatures equal to 1.
  An easy calculation gives the following simple expressions
  for $A$ and $B\,:$
   \begin{equation}
  A=K_1^{-1/3}K_2^{-2/3}\,,\qquad  B=K_1^{-2/3}K_2^{-1/3}\,.
  \label{AB}
  \end{equation}
  In our case $K_1=K_2=4\pi^2g\,,$ and, using the above scaling,
  we can expect that, after adding few iterates, the size of 
 the central window which contains the central island reduces
to some amount which is of order 1 in the scaled variables
$\xi,\eta,$ or of the order $1/g$ in the old variables $x,y\,.$
Denote this reduced central window by $w^*\,.$ Consider the image,
$F(w^*)\,,$ of the reduced window which looks like a horseshoe.
 It lies somewhere in the horizontal string of FLE(1), the $x$-coordinate
near $g {\rm mod}\frac12\,,$ and moves to the right with velocity close to 1 if 
one increases the parameter $g\,.$ Since the $sin(2\pi x)$ is close
to zero in $w^*\,,$ the map does not change strongly the sizes of
the cell, and we conclude that the width and the height of 
the horseshoe, $F(w^*)\,,$ is also of  the order $1/g\,.$
The necessary condition for
the existing of the central island reads now as:
\begin{equation}
F(w^*)\cap w^*\ne\emptyset
\label{condition1}
\end{equation}
 In view of the above description of the position, the sizes,
 and the velocity of
$F(w^*)\,,$ one obtains easily that (\ref{condition1}) holds
if the parameter $g$ runs trough an interval of the width of
the order $1/g\,,$ placed in the vicinity of $N/2\,,$
where $N$ is an integer,   and the sizes of the central island in
its maximal phase are also  of the order $1/g$ in both coordinates.
Both numerical experiments and analytic calculations (see \cite{LPS})
show that this result is in a very good accordance with the
reality. It also follows from \cite{LPS} that the condition 
(\ref{condition1}) is close to the sufficient one.
\medskip

Consider now the question whether there exist islands which are
inside other cells of FLE(1) and such that 
their iterates never quit FLE(1). It is convenient to supply them
with some labels, or codes. To this end, let us construct a
symbolic representation for the dynamical system $(F,Z)$ taking
a partition ${\cal P}$ of $Z$ created by the cells of the web 
$W_0^u\cup W_0^s\cup W_1^s\,.$ These lines together with all 
its images with respect to  $F^k\,, k=0,\pm 1,\pm 2,\dots,$
constitute a set of Lebesgue measure zero. So we can restrict
ourselves with considering points whose trajectories never
fall on the boundaries of the elements of ${\cal P}\,.$
Denote by $N$ the number of the elements of ${\cal P}\,.$
The following asymptotic estimate is true
for  large values of $g\,:$ $N\sim 4g\,.$
Let us attach numbers from 1 to $N$ to the elements of
the partition ${\cal P}$ in some order: 
${\cal P}=\bigcup_{i=1}^{N}A_i\,.$
Then almost each point $z$ of $Z$ acquires an infinite code
$(\dots,n_{-2}(z),n_{-1}(z),n_{0}(z),n_1(z),n_2(z),\dots)$ by the  rule:
\begin{equation}
F^i(z)\in A_{n_i(z)}\,, i\in {\bf Z}\,. 
\label{symbdyn}
\end{equation}
The equality (\ref{symbdyn}) defines a sequence of functions
$n_i\,:Z\to \{1,2,\dots,N\}\,, i\in{\bf Z}\,.$   

We supply a non-central cell, $c\,,$ of the horizontal part of
FLE(1) with a {\it code} which consists of one number
equal to $n_0(z)\,, z\in c\,, $ and a non-central cell,
$c^{\prime}\,,$ belonging to the vertical part of FLE(1)
with the 1-letter code equal to $n_{-1}(z)\,, z\in c^{\prime}\,.$
It is easy to verify that the cells symmetric with respect to 
the diagonal
 possess the same code. Note that the exceptional cells
marked by letter E on Fig.1 acquire a same code which is in
the accordance with the mentioned above gluing of
the adjacent couples of the exceptional cells together.

Suppose now that a non-central cell $c$ belonging to the horizontal
part of FLE(1) with code $k$ contains an island whose images never 
quit FLE(1). 
Then the first forward image of the island should be
inside the non-central cell $c^{\prime}$ belonging to the vertical
part of FLE(1) with the same code as $c\,,$ that is symmetric
to $c\,.$ As well as in the case of the central island, the
 the spot
containing the island appears to be much smaller than  the
cell $c\,.$ Let us apply the same heuristic analysis as in
the case of the central island by use of families of parabolas.
The horizontal family of the parabolas in our case is the same
as in the case of the central island, and their maximal curvatures
have the same order, that is $K_1\sim g\,.$ To obtain the vertical
family of parabolas in the cell $c$ one has to take the preimage
of that for the central island via the map $F\,.$ Since the linear
part of the map $F$ is close to the diagonal linear map with
the matrix $\left(\begin{array}{cc}\lambda & 0\\0& \lambda^{-1}
\end{array}\right)$ with $\lambda\sim g\,,$ the maximal curvature
of the preimage of a parabola from the initial vertical
family acquires a multiplier of the order $\lambda^3\,.$
So, the vertical family of parabolas passing trough the cell $c$
has the maximal curvature, $K_2\,,$ of the order $g^4\,.$
Substituting these estimates for $K_1$ and $K_2$ into (\ref{AB}),
we obtain the following estimates for the sizes of the window,
denote it by $w^*_k$ where $k$ is the code of $c\,,$ where 
the island may reside if it exists in the cell $c$ (recall
that in this section we study only the islands whose iterates never
quit FLE(1)):
\begin{equation}
{\rm the \ width\ of}\ w^*_k\sim g^{-3}\,,\ \ 
{\rm the \ height\ of}\ w^*_k\sim g^{-2}\,,
\label{sizes}
\end{equation} 
Let now determine the position and the size of the interval in
the real line of the parameter $g$ where an island of such kind
can exist. The image of $w^*_k$ under the map $F$ coincides
essentially with the symmetric window, $w^{\prime *}_k\,,$
on the vertical part of FLE(1). The second image of $w^*_k\,,$
that is $w^{\prime\prime*}_k=F(w^{\prime *}_k)=F^2(w^*_k)\,,$ 
is a small horseshoe which lies
again in the horizontal part of FLE(1). This can be done
starting with each cell $c$ in the horizontal part of FLE(1).
Therefore, on this horizontal part we have plenty of
windows $\{w^*_k\}$ and horseshoes $\{w^{\prime\prime*}_k\}$
the index $k$ ranging all but one possible values.
They create two more or less regular sequences of spots where 
island can exist. If there exists an island in $w^{\prime\prime*}_k$
it must be contained in some initial window, say $w^*_l\,.$
This gives a necessary condition for the existing of an island
of the considered type:
\begin{equation}
w^{\prime\prime*}_k\cap w^*_l\ne\emptyset\,.
\label{condition2}
\end{equation}
If $k=l$ in this relation, we expect an island of period 2.
This means that all KAM circles which contain in $w^*_k$
move under the first iterate to $w^{\prime *}_k\,,$ and then,
after applying the next iterate return to  $w^*_k\,.$
The periods of KAM curves in this case have to be even.
 If $k\ne l\,,$ we obtain an island of period 4.

To estimate the width of the interval in the real axis of the parameter
$g$ containing the values of $g$ for which the relation
(\ref{condition2})
takes place, one has to take into account that the string of 
initial small windows, $ \{w^*_k\}\,,$ undergoes little changes
when one starts to  change $g\,,$ while the string of horseshoes,
  $\{w^{\prime\prime*}_k\}\,,$ starts to move along FLE(1)
with a velocity close to 1. So, the width of the interval
in $g$-axis where the collision (\ref{condition2}) occurs 
is of the order $g^{-3}\,.$ This estimate together with the
estimates (\ref{sizes}) for the sizes of the window containing
the island both are in an excellent agreement with analytic
an numerical analysis made for such kind of islands in
\cite{LPS}. 
Let us call the considered islands {\it the islands of the order 1.}
Denote by ${\cal E}_1$ the set on the axis of $g$ consisting
of the values with the property that the corresponding to
this value map has an island of the order 1. 
Since the number of the collisions of the form (\ref{condition2})
when $g$ ranges the interval $[n/2,(n+1)/2]\,,\, n$ integer,  
equals to the product of the number of the windows and the number
of the horseshoes, we obtain that the total estimate of the Lebesgue
measure of ${\cal E}_1\cap [n/2,(n+1)/2]\,:$
\begin{equation}
leb({\cal E}_1\cap [n/2,(n+1)/2])\le {\rm const}\, n^{-1}\,.
\end{equation}

\section{FLE's and islands of higher orders}

We would like to construct a decreasing sequence FLE(n), n=1,2,3,...,
such that 1)FLE(n) would consist of the cells created by the web
$\bigcup_{k=0}^n\left(W_k^u\cup W_k^s\right)\,;$ 2)it would
as simple as possible, in particularly, it would consist of
two parts: vertical an horizontal trunks, and the cells of
each trunk would be ordered in a linear way following to
a more or less smooth curve. 

We can try to construct the sequence of FLE's inductively.
Suppose all FLE(k) with $k<n$ are constructed.
To build FLE(n) let us first subdivide the cells of FLE(n-1)
by the all possible arcs of $W_n^u$ and $W_n^s\,.$ The
second step is to subject the new smaller cells inside
FLE(n-1) to a selection by the ''geometric test''
described in the Section 3. The union of the remaining
cells is still too complicated and form a kind of
a ''tree'' and a more complicated structure. To select
the ''main trunk'' one should ''cut off'' the 
''lateral branches''. The motivation is that these
lateral branches can be filled by the images of parts
of the main trunk and some smaller good cells. 
The following hint can be useful in selecting the cells
belonging  to FLE(n). Consider the $n$-th iterates
of the fields of the stable and the unstable directions
defined at the end of Section 3: $X_n^u=F^n_*(X^u)$ and
$X_n^s=F^{-n}_*(X^s)$ and denote by $D_n(z)$ the square of
the sinus of the angle between these two directions at
a point $z\,.$ Then the above ''geometric''  selection
is the same as to retain the cells containing zeroes
of  $D_n(z)\,.$ The latter constitute a smooth curve.
Let us consider the {\it curvatures} of the lines of
the fields $X_n^u$ and  $X_n^s$ passing through a point
of this curve and take the maximal of these two. The
curvature so defined can help to select lateral branches
from the main trunk: at points of the lateral line
the value of the curvature is much bigger than at points
of the zero line passing trough the main trunk.
Unfortunately, at the present moment, there is no
rigorous description of the construction in question,as well
as a rigorous proof that it provides a FLE. This will be
a subject of future investigations.

Suppose we have constructed FLE(n). Then the same analysis
as in preceding section can be carried out to determine if
there exist an island inside a cell $c$ of FLE(n).
Suppose that $c$ belongs to the horizontal trunk of FLE(n). 
As
well as in the case of FLE(1), the cell $c$ acquires a
code $k$ which is now a string of $n$ symbols:
$k=(k_1,k_2,\dots,k_n)$ where $k_i=n_{i-1}(z)\,, z\in c$
(recall that  the functions $n_k(z)$ are defined by the relation
(\ref{symbdyn})). 
If the cell $c$ contains an island then
its $n-1$ iterates are outside of FLE(n) and the $n$-th
iterate falls into a cell, $c^{\prime}\,,$ belonging 
to the vertical trunk of FLE(n). The code of $c^{\prime}$
is reverse to $k\,,$ it is 
$\overline{k}=(k_n,k_{n-1},\dots,k_1)\,.$ Then the same analysis
yields the following necessary condition for the existence
of an island inside $c\,.$ This island is confined in a much
smaller window $w^*_k\subset c\,.$ 
Denote 
$w^{\prime *}_{\overline{k}}=F^n(w^*_k$ and 
$w^{\prime\prime *}_{\overline{k}}=F(w^{\prime *}_{\overline{k}})$
(a horseshoe). Then the condition is 
\begin{equation}
w^{\prime\prime*}_{\overline{k}}\cap w^*_l\ne\emptyset
\label{condition3}
\end{equation}   
for some window $ w^*_l$ belonging to the horizontal trunk of
FLE(n). To calculate  sizes of a typical window $w^*_k$ one
should take $K_1\sim g$ and $K_2\sim g^{1+3n}$ in 
(\ref{AB}) since the
second (vertical) family of parabolas is a reverse image of
the initial vertical family with maximal curvature of order $g$
via the $n$-th iterate of $F\,,$ each iterates supplying a multiplier
of the order $g^3\,.$ Substituting this into (\ref{AB}) gives 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} 
and in its turn the estimate of the order $g^{-1-2n}$ for
the width of the interval of the values of the parameter $g$
for which the collision (\ref{condition3}) takes place.
Putting together the contributions coming from the
collisions between all possible horseshoes, 
$w^{\prime\prime*}_{\overline{k}}\,,$ and targets, $w^*_l\,,$
one gets the following estimate for the density of the set
${\cal E}_n$ of the values of the parameter $g$ for which
there exists an island of the considered type (let us
call them islands of the order $n$):
\begin{equation}
leb({\cal E}_n\cap [p/2,(p+1)/2])\le {\rm const}\, p^{-1}\,.
\label{badestimate}
\end{equation}
This looks desperating: by throwing out a relatively small
set we can kill islands of any finite order, but cannot
kill all of them! But in fact the situation is a little bit
better. In the case of the central island and the islands
of the order 1 {\it every} collision of a ''horseshoe''
with a ''target window'' resulted in the creating of
an island, as it follows from \cite{LPS}. The situation
is different for the island of higher orders. Indeed,
the relation (\ref{condition3}) is necessary for the the
existing of an island inside $w^*_k\,.$ It means that
the island inside $w^*_k$ comes after $(n+1)$-th iterate
to another window, $w^*_l\,.$ But the same condition
should be fulfilled also for $w^*_l\,:$ otherwise it could
not contain an island. In fact, the trajectory of an island
should be periodic, and hence this chain of collisions should
be closed. So the analysis which counts {\it all} collisions
of the form (\ref{condition3}) is superfluous. Let us call
the collision (\ref{condition3}) ''good'' if the corresponding
horseshoe of the target, namely 
$w^{\prime\prime *}_{\overline{l}}\,,$ has no collision with
some $w^*_s\,.$ If a collision is good, then it cannot
produce an island. This classification can reduce strongly
the possible cases of the birth of islands. We have a string
of targets, $\{w^*_l\}$ and another string of horseshoes,
$\{w^{\prime\prime *}_{\overline{k}}\}\,$ the latter moving
along the former with velocity of the order 1 when one moves
$g\,.$ A kind of interferention of these two (in some sense
regular) strings will produce islands, and a quasiperiodicity
could help to avoid the most double collisions (cf. \cite{Laz2}).  
\medskip

Nevertheless there are codes for which the collision of the form
(\ref{condition3}) always produces an island. There are two 
cases of special codes:

I. Both $k$ and $l$ are {\it symmetric, } that is $k=\overline{k}$
and $l=\overline{l}$ where the dash over a string denotes the 
inversion of the string: $\overline{(k_1,k_2,\dots,k_n)}=
(k_n,k_{n-1},\dots,k_1)\,.$ In this case the island in the 
window $w^*_k$ comes
after $n$-th iterates to a symmetric window $w^{\prime *}_k\,,$
then, after the next iterate it falls, due to (\ref{condition3}),
in $w_l^*\,,$ and after additional $n$ iterates it comes to
a symmetric window $w^{\prime *}_l$ on the vertical trunk.
Since all figure is symmetric with respect to diagonal
(or the same: with respect to the action of the reversor $R$),
the window $w^{\prime *}_l$ meets a horseshoe which is
symmetric to $w^{\prime\prime*}_{k}$ and, again by the symmetry
(recall that the reversor conjugates $F$ with $F^{-1})\,,$
this new horseshoe is the preimage of the initial window $w_k\,.$
Therefore the trajectory of the island trough the windows has 
period $2n+2$ if $k\ne l\,.$ The number of such ''symmetric''
collisions
is estimated from above by ${\rm const}\, g^n\,.$

II. The case $k=l\,.$ This cases produces an island of period
$n+1\,.$ The number of such collisions is again of the order
$g^n\,.$ 
      
We conclude the paper by formulating the following

\begin{conjecture}
In the case of the standard map  there exist 
only the islands
listed above in the  cases I and II. 
\nonumber
\end{conjecture}

If this conjecture is true then the
right hand side in the  estimate (\ref{badestimate})
can be replaced by $\left({\rm const}\, p\right)^{-n-1}\,.$
Since all the estimates for the individual windows come from
some kind of renormalization procedure, self-similarity reasons
involve that the constant in this estimate can be chosen 
independent of $n$ and $p\,.$ By our construction, all island
belong to some cells in FLE's and hence 
${\cal E}=\bigcup_{k=0}^{\infty}{\cal E}_k$ where ${\cal E}_0$
is the set of values of the parameter $g$ for which the
central islands exists, ${\cal E}_n\,,n\ge 1\,,$ is the set
of parameters corresponding to existing islands of the order $n\,.$
Putting all these estimate together gives us the final estimate
for the density of ${\cal E}\,:$
$$
leb({\cal E}\cap [p, p+1])\le {\rm const}\, p^{-1}\,.
$$   




\section{Acknowledgments}  
 This investigations was partially supported
by INTAS grant 97-0771, and for the second author by RFFI grant
97-01-00612 and the grant of the State Higher Educational
Committee in Russia.

The paper was written during the visit of the second author in
July 1999 at the University of Milano Bicocca. The visit has been
supported by INDAM, Gruppo Nazionale per la Fisica Matematica.

%\newpage

\vskip 0.5 truecm

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