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\title{The standard map and a probabilistic analogue}
\author{L.D. Pustyl'nikov}
\date{ }
\maketitle
\begin{abstract}
This paper is devoted to the study of
statistical properties of trajectories of a class of
dynamical systems which includes the standard map. A
well known conjecture is formulated and its natural
probabilistic analogue is defined in the framework of
ergodic theory of random transformations.
The main results is the proof of the probabilistic analogue
of this conjecture.
\end{abstract}
Let $\varphi = \varphi(u)$ be a smooth function having the period
1 with respect to $u$, $\varphi (u) = \varphi (u+1)$. We consider
the map $A_\varphi : (u,z) \to (u^\prime, z^\prime)$ of the cylinder
$C = \{ u, z : 0 \le u < 1 , - \infty < z < \infty\}$ having
standard type
\begin{equation}
A_\varphi : \left\{ \begin{array}{l l}
u^\prime = u + z^\prime \ \mbox{mod}\ 1\\
z^\prime = z + \varphi (u) \end{array} \right. ,
\end{equation}
which in the special case $\varphi = \varphi_\ast (u) = h \sin (2\pi
u)$
is the standard map
\begin{equation}
A_\ast : \left\{ \begin{array}{ll}
u^\prime = u + z^\prime \ \mbox{mod}\ 1 & \\
z^\prime = z + h \sin (2\pi u) & \end{array}
\right. .
\end{equation}
The maps $A_\varphi$, and especially $A_\ast$ were studied in many
papers
(see [1] - [7]), and some conjectures connected with these maps were
formulated and are very popular at the present time.
Conjecture. For sufficiently large value of the parameter $h$ there
exists a set $Q \subset C$ having positive Lebesgue measure such that
if $z_n = z_n (u,v)$ is the coordinate of the point
$(u_n, z_n) = A^n_\ast (u,z)$ ($A_\ast^n$ is the n-th power of
$A_\ast$)
then the quality
$I_n = \int_Q z^2_n (u,z) du dz$ behaves like $cn$ as $n \to \infty$,
where the constant $c \not= 0$.
The sense of this conjecture is that for initial data from $Q$ the
change of coordinate $z_n$ behaves like a trajectory of random walk
and
as indicated in [7] at the present time we are very far from its
proof.
Originally this conjecture assumed that the Lebesgue measure of
complement
to sets $Q$ on the cylinder $C$ is equal to zero, and for this case
it
was disproved in the paper [5], where it was proved that on the
half-line $h > 0$ there exists an open set $H$ having infinite
Lebesgue
measure on the straight line, such that for any $h \in H$ there
exists an open set $\Omega \subset C$ for which the quality $I_n$
behaves like $cn^2$ as $n\to \infty$, where a constant $c \not= 0$.
The aim of present paper is to formulate and to prove a probabilistic
analogue of the mentioned conjecture in the framework of ergodic
theory of random transformations ([9], [10]). The essence of our
approach
is that instead of iterations of map $A_\ast$ we consider consecutive
actions of two maps $A_\ast$ and $A_\varphi$ with
$\varphi \not= \varphi_\ast$ so that with probability
$\frac{1}{2}$ we apply one of these maps. If the functions
$\varphi_\ast$ and $\varphi$ are close in some metric sense then it
is possible to say that such random dynamical system approximates
original determinated system.
Let us introduce the following objects:
\noindent
$T = \{ u,z : 0 \le u < 1, 0 \le z < 1\}$ is the flat two-dimensional
torus;
\noindent
$S = \{ 0,1\}$ is the set, consisting of two elements: $0$ and $1$;
\noindent
$\nu$ is the measure on $S$ such that measure of each element is
equal to
$\frac{1}{2}$;
\noindent
$\Omega$ is the space, that is the direct product of countable
number of sets $S$, and an arbitrary point of $\Omega$ is one-sided
sequence $\omega = (\omega_1, \omega_2, \ldots)$ in which for any
$n = 1, 2, \ldots$ the quantity $\omega_n$ takes one of two values:
$0$ or $1$;
\noindent
$p$ is Bernulli measure on $\Omega$, that is the countable product
of measures $\nu$;
\noindent
$\sigma$ is the map of $\Omega$ into itself, such that if $\omega =
(\omega_1, \omega_2, \ldots) \in \Omega$, then $\sigma (\omega) =
\omega^\prime = (\omega^\prime_1, \omega^\prime_2, \ldots)$, where
$\omega_n^\prime = \omega_{n+1}$ $(n = 1, 2, \ldots)$;
\noindent
$W = C \times \Omega$ and $\hat{W} = T\times \Omega$ are the direct
products of the spaces $C$ and $\Omega$ and the spaces $T$ and
$\Omega$
respectively;
\noindent
$\mu$ is the measure on $W$ and $\hat{W}$, being the product of the
Lebesgue measure on $C$ and the measure $p : d\mu = dudz \times dp$;
\noindent
$\psi (u,\omega)$ is the function on $W$ having the form
\[
\psi (u, \omega) = \left\{ \begin{array}{l l l}
\varphi_\ast (u) , & \mbox{if} & \omega_1 = 0\\
\varphi (u) , &if & \omega_1 = 1 \end{array} \right. ;
\]
\noindent
$B$ is the map of the space $W$ into itself such that if
$(u,z,\omega) \in
W$ then $B(u,z,\omega) = (u^\prime, z^\prime, \omega^\prime)$, where
$(u^\prime, z^\prime) = A_{\psi(u,\omega)}(u,z)$, $\omega^\prime =
\sigma
(\omega)$.
\bigskip
\noindent
Definition 1. The map $B$ of the space $W$ is called the
probabilistic
analogue of the standard map $A_\ast$ of the cylinder $C$.
\bigskip
\noindent
Theorem 1. There exists the measurable and integrable with respect
to measure $\mu$ function $f(u,z,\omega)$ on $\hat{W}$ such that
\[
\int_{\hat{W}} f(u,z,\omega) d\mu = \frac{1}{2} \int^1_0 \varphi (u)
du
\]
and for almost all points $(u,z,\omega) \in W$ (the measure $\mu$ of
the complement is equal to zero) the coordinate $z^{(n)} (n =
1,2,\ldots)$
of points $(u^{(n)}, z^{(n)}, \omega^{(n)}) = B^n (u,z,\omega)$
satisfy
the equality
\[
\lim_{n\to\infty} \frac{z^{(n)}}{n} = f(u,\{ z \}, \omega),
\]
where $\{ z \}$ is the fractional part of $z$.
\bigskip
\noindent
Proof. We consider the function $\hat{\psi} (u,z,\omega) = \psi
(u,\omega)$
on $\hat{W}$ and map $\hat{B}$ of the space $\hat{W}$ having the
following
form: if $(u,z,\omega) \in \hat{W}$ then $\hat{B} (u,z,\omega) =
(\hat{u}, \hat{z}, \hat{\omega})$, where $\hat{u} = u+\hat{z} mod 1$,
$\hat{z} = z + \hat{\psi} (u,z,\omega) mod 1$, $\hat{\omega} = \sigma
(\omega)$. By virtue of the definition of the measure $\mu$,
the maps $B$, $\hat{B}$ and the conditions of theorem 1 $\mu
(\hat{W}) = 1$,
the map $\hat{B}$ preserves the measure $\mu$ on $\hat{W}$ (that is
measure of a set is equal to the measure of it's inverse image) and
for any $n = 1,2,\ldots$ the equality $z^{(n)} = \sum^{n-1}_{k=0}
\hat{\psi} (\hat{B} (u,z,\omega))$ holds. Therefore the assertion
of theorem 1 follows from the Birkhoff-Khinchin ergodic theorem
applied to the function $\hat{\psi} (u,z,\omega)$ and map $\hat{B}$
of the space $\hat{W}$ with the measure $\mu$.
Theorem 1 is proved.
\bigskip
\noindent
Definition 2. We introduce the probabilistic analogue of the
conjecture:
for sufficiently large $h$ in the definition of the map $A_\ast$
there exists a set $\Pi \subset C$ having a positive and finite
Lebesgue
measure on $C$ such that if we consider the coordinate $z^{(n)} =
z^{(n)} (u,z,\omega)$ of point $(u^{(n)}, z^{(n)}, \omega^{(n)}) =
B^n (u,z,\omega)$ then the quantity ${\cal J}_n =
\int_{\Pi\times\Omega}
(z^{(n)} (u,z,\omega))^2 d\mu$ behaves like $cn$ as $n \to \infty$,
where a constant $c \not= 0$.
Further we shall formulate and prove the theorem 2, from which it
follows that on the half-line $h > 0$ there exists an open set $H$
having infinite Lebesgue measure on the straight line such that for
any $h \in H$ there exists a smooth function $\varphi(u)$ having the
period 1 and differing from $\varphi_\ast (u)$ on the interval of
arbitrarily small measure and conciding with $\varphi_\ast$ for other
points, such that there exists an open set $\Pi \subset C$ for which
the probabilistic analog of conjecture holds.
In addition there exist infinitely-many such sets on $C$ having the
same measure which are mutually disjoint. If the function
$\varphi(u)$
coinsides $\varphi_\ast (u)$ then the probabilistic analog of
conjecture
coinsides the conjecture. The difference of these function on the set
of
small measure is a natural approximation from probabilistic point of
view.
At the end of the paper a corollary is formulated in which states
that
the random map $\hat{B}$ introducing in the proof of theorem 1
is not ergodic. This corollary shows that the result of theorem 2
is not trivial because the general methods of probability theory
based on using of martingale theory and central limit theorem are
not applicable in this case.
\bigskip
\noindent
Theorem 2. On the half-line $h>0$ there exists an open set $H$ having
infinite Lebesgue measure on the straight line such that if $h \in H$
then there exists an integer $k=k(k) > 0$ such that the following
assertion holds: for any $\epsilon$ satisfying the inequality
$0 < \epsilon < 1 $ there exist numbers $\alpha, \beta$, infinitely
differentiable function $\varphi(u)$ and an infinite number of
mutually disjoint open sets $\Pi_m \subset C$ $(m\in {\bf Z})$ such
that
1) $0 < \alpha < \beta < 1, \beta - \alpha \le \epsilon$;
2) if $u \in [0,\alpha] u[\beta,1)$ then $\varphi(u) = \varphi_\ast
(u)$
and if $u \in (\alpha, \beta)$ then $|\varphi (u) - \varphi_\ast
(u)| \le 2k$;
3) for any $m \in {\bf Z}$ Lebesgue measure of set $\Pi_m$ is equal
to $\kappa = \int_{\Pi_0} dudz$, where $0 < \kappa < 1$, and the
quantities ${\cal J}^{(m)}_n = \int_{\Pi_m \times \Omega} (z^{(n)}
(u,z,\omega))^2 d\mu$ satisfy the equality
\begin{equation}
\lim_{n\to\infty} \frac{{\cal J}^{(m)}_n}{n} = k^2 \kappa \ ,
\end{equation}
(the quantities $z^{(n)} (u,z,\omega)$ are taken from definition 2).
\bigskip
\noindent
Proof. Suppose that there exists $u_0$ satisfying the inequality
$0 < u_0 < 1$, and integer number $k$ such that
\begin{equation}
h \sin (2\pi u_0) = k > 0\ ,
\end{equation}
\begin{equation}
- 4 < 2\pi h \cos (2\pi u_0) = \rho < 0 \ ,
\end{equation}
\begin{equation}
\rho \not= - 2 + 2 \cos \frac{2\pi s}{q}\ , \quad (s = 0, \pm 1,
\ldots , \pm q; q = 1,2,3,4)\ ,
\end{equation}
and introduce the quantities
\begin{equation}
z^{(0)}_n = nk \ , 3n \in {\bf Z}\ .
\end{equation}
By virtue of (2), (4) and (7) we have quality
\begin{equation}
A_\ast (u_0, z^{(0)}_n) = (u_0, z^{(0)}_{n+1}), n \in {\bf Z}\ .
\end{equation}
Let
\begin{equation}
\delta = \frac{1}{2} \min (\epsilon, u_0, 1-u_0)\ ,
\end{equation}
and define numbers
\begin{equation}
\alpha = u_0 - \delta\ , \beta = u_0 + \delta\ , \alpha^\prime = u_0 -
\frac{\delta}{2}\ , \beta^\prime = u_0 + \frac{\delta}{2}\ ,
\end{equation}
which by virtue of (9) satisfy the inequalities
\begin{equation}
0 < \alpha < \alpha^\prime < u_0 < \beta^\prime < \beta < 1\ , \beta
< \alpha \le \epsilon\ .
\end{equation}
We consider the infinite-differentiable function $\varphi =
\varphi(u)$
having period 1 with respect $u$ such that the following condition
hold:
\begin{equation}
\varphi (u) = \varphi_\ast (u) \qquad \mbox{if} \qquad u \in
[0,\alpha]
\bigcup [\beta,1)\ ,
\end{equation}
\begin{equation}
\varphi (u) = \varphi_\ast (u) - 2 k \qquad \mbox{if} \qquad u \in
[\alpha^\prime, \beta^\prime]\ ,
\end{equation}
\begin{equation}
| \varphi (u) - \varphi_\ast (u) | \le 2 k \ , \qquad u \in [0,1)\ .
\end{equation}
According to (4) and (13) $\varphi (u_0) = - k$, and therefore by
virtue of (1) and (7) we obtain:
\begin{equation}
A_\varphi (u_0, z^{(0)}_n ) = (u_0, z^{(0)}_{n-1})\ , n \in {\bf Z}\ .
\end{equation}
For any $n \in {\bf Z}$ we introduce two sequences of maps
\begin{equation}
A^{(n)} : (u,z) \to (u^\ast, z^\ast) = A_\ast (u + u_0 , z +
z^{(0)}_{n-1})
- A_\ast (u_0, z^{(0)}_{n-1}),
\end{equation}
\begin{equation}
\hat{A}^{(n)} : (u,z) \to (\hat{u}^\ast, \hat{z}^\ast) = A_\varphi
(u + u_0, z+z^{(0)}_{n+1}) - A_\varphi (u_0, z^{(0)}_{n+1}),
\end{equation}
and remark that from the equalities (1), (2), (13), (16) and (17)
it follows that if $\alpha^\prime - u_0 \le u \le \beta^\prime - u_0$
then for any $n \in {\bf Z}$ maps $A^{(n)}$ and $\hat{A}^{(n)}$ are
the
same as the map $\hat{A} : (u,z) \to (\hat{u}, \hat{z})$ having the
form
\begin{equation}
\hat{A} : \left\{ \begin{array}{l}
\hat{u} = u + \hat{z}\\
\hat{z} = z + h \sin (2\pi (u_0 +u)) - h\sin (2\pi u_0)
\end{array}\right.
\end{equation}
and $(0,0)$ is the fixed point. The map $\hat{A}$ preserves the area
and by virute of (5) the trace $Sp d\hat{A}$ of matrix
\[
d\hat{A} (0,0) = {1 + 2\pi h \cos (2\pi u_0) \qquad 1 \choose 2 \pi
\cos
(2\pi u_0) \qquad ~~~1}\ ,
\]
defining the linear part of $\hat{A}$ at the point $(0,0)$, satisfies
the inequality $|Sp d\hat{A} | < 2$. Therefore the point $(0,0)$ is
of
elliptical type. Further we shall use the results of paper [5]
(theorem 1
and its proof) from which it follows that on the half-line $h > 0$
there
exists an open set $H$, having infinite Lebesgue measure on the
straight
line, such that for any $h \in H$ there exists $u_0$ and integer $k$
such
that the equations (4), (5), (6) hold and the fix point $(0,0)$ of
the
map $\hat{A}$ is a point of general elliptical type, i.e. in its
normal
Birkhoff form (see [11]) $\theta^\prime = \theta + a_0 + a_1 \tau +
\ldots, \tau^\prime = \tau$ ($\theta,\tau$ are certain polar
coordinates;
$\theta$ is angle, $\tau$ is radius) the coefficient $a_1 \not= 0$.
Therefore if $h \in H$ then Moser's theorem [12] can be applied to
the
map $\hat{A}$. According to this theorem in any neighborhood of the
point
$(0,0)$ there exists a continuous closed curve arounding the point
$(0,0)$
and invariant with respect to $\hat{A}$. Let us take curve $\gamma$
such that the points $(\tilde{u}, \tilde{z})$ lying inside the domain
$\Gamma$ bounded by this curve, satisfy the inequalities
\begin{equation}
\alpha^\prime - u_0 < \tilde{u} < \beta^\prime - u_0\ , 2|\tilde{z}|
< \epsilon \ ,
\end{equation}
and for any $m \in {\bf Z}$ we define the domain $\Pi_m \subset C$
of the following form:
\begin{equation}
\Pi_m = \{ u,z : u = \tilde{u} + u_0 \ , z = \tilde{z} + z_m^{(0)}\ ,
(\tilde{u}, \tilde{z}) \in \Gamma \} \ ,
\end{equation}
where $z^{(0)}_m$ are the quantities introduced in (7) for $m = n$.
>From the invariance of the curve $\gamma$ with respect to $\hat{A}$
by
virtue of (20), (19), (13), (7), (17) and definitions of maps
$\hat{A},
A^{(n)}, \hat{A}^{(n)}$ in (18), (16) and (17) for any $m \in {\bf
Z}$
we obtain the relations
\begin{equation}
A_\ast (\Pi_m) = \Pi_{m+1}\ , A_\varphi (\Pi_m) = \Pi_{m-1}\ ,
\end{equation}
and according to (19), (20) and (7) the coordinate $z_m$ of
any point $(u_m, z_m) \in \Pi_m$ satisfies the inequality
\begin{equation}
| z_m - mk | < \frac{\epsilon}{2}\ .
\end{equation}
We prove now that the domains $\Pi_m (m\in {\bf Z})$ satisfy the
assertions of theorem 2. By virtue of (22) for small $\epsilon$
the domains $\Pi_m$ don't intersect each other for distinct $n$.
Let $(u,z) \in \Pi_m$, $\omega \in \Omega$ and for all natural $n$
we consider points
$(u^{(n)}, z^{(n)}, \omega^{(n)}) = B^n (u,z,\omega)$.
According to (21) and definition of $B$ there exists integer
$m_n$ such that the point
\begin{equation}
(u^n, z^{(n)}) \in \Pi_{m_n}\ .
\end{equation}
Therefore introducing the value $z^{(n)}$ by the formula
$z^{(n)} = k m_n + \xi_n$ by virtue of (23), (22), definition 2
and definition of the value ${\cal J}^{(m)}_n$ in the formulation
of theorem 2 we have
\begin{equation}
|\xi_n | < \frac{\epsilon}{2}\ ,
\end{equation}
\begin{equation}
{\cal J}^{(m)}_n = {\cal J}^{(m)}_{n,1} + {\cal J}^{(m)}_{n,2} +
{\cal J}^{(m)}_{n,3} + {\cal J}^{(m)}_{n,4} \ ,
\end{equation}
where
\[
{\cal J}^{(m)}_{n,1} = k^2 \int_{\Pi_m \times \Omega} (m_n - m)^2
d\mu,
{\cal J}^{(m)}_{n,2} = 2k \int_{\Pi_m \times \Omega} (m_n - m)
\xi_n d\mu\ ,
\]
\[
{\cal J}^{(m)}_{n,3} = 2k^2 m\int_{\Pi_m \times \Omega} (m_n - m)
d\mu, {\cal J}^{(m)}_{n,4} = \int_{\Pi_m \times \Omega} (km +\xi_n)^2
d\mu\ .
\]
Let us estimate each of the integrals ${\cal J}^{(m)}_{n,s}$
($s = 1,2,3,4$). To calculate the integrals ${\cal J}^{(m)}_{n,1}$
and ${\cal J}^{(m)}_{n,3}$ we note, that the quantity $\eta_n = m_n -
m$
is the realization of the sum $\sum^n_{i=1} x_i$ of $n$ independent
random values $x_1, \ldots, x_n$ taking on any of two values
$x_i = \pm 1 \ \ (i=1,\ldots,n)$ with probability $\frac{1}{2}$,
and therefore according to well known statements of probability
theory ([8]) we have the equalities $\int_\Omega (m_n - m)^2 dp = n$,
$\int_\Omega (m_n - m) dp = 0$. Substituting these equalities into
integrals ${\cal J}^{(m)}_{n,1}$ and ${\cal J}^{(m)}_{n,3}$ we obtain
the following relations:
\begin{equation}
{\cal J}^{(m)}_{n,1} = k^2 n \int_{\Pi_m} du dz \ ,
{\cal J}^{(m)}_{n,3} = 0\ .
\end{equation}
By virtue of the independence of $m_n - m$ and $\xi_n$
\begin{equation}
{\cal J}^{(m)}_{n,2} = 0\ ,
\end{equation}
and the inequality
\begin{equation}
|{\cal J}^{(m)}_{n,4} | < (km + \epsilon)^2 \int_{\Pi_m} du dz
\end{equation}
are valid. According to (2), (21) and the formulation of
theorem 2 the area $\int_{\Pi_m} du dz$ of the domain $\Pi_m$ on
the cylinder $C$ does not depend on $m$ and is equal to $\kappa$.
Then substituting (26), (27) and (28) into (25) we obtain (3).
Theorem 2 is proved.
\bigskip
\noindent
Corollary. If $H$ is the set constructed in theorem 2 and $h \in H$,
then the map $\hat{B}$ of the space $\hat{W}$ constructed in the
proof
of theorem 1, is not ergodic.
The corollary follwos from the proof of theorem 2 since the set
$\Pi_0 \times\Omega$, constructed there is invariant with respect
to $\hat{B}$.
The result of theorem 2 in which instead of asymptotic equality (3)
there are asymptotic inequalities was obtained in [6].
\bigskip
\noindent
Acknowledgements. The author thanks BiBoS Research Center for the
support.
\newpage
\noindent
{\bf References}
\bigskip
\noindent
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\end{description}
\end{document}