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\begin{document}


\issueinfo{10}{13}{}{2004}
\dateposted{October 26, 2004}
\pagespan{113}{121}
\PII{S 1079-6762(04)00136-2}
\copyrightinfo{2004}{American Mathematical Society}
\revertcopyright
\commby{Svetlana Katok}
\subjclass[2000]{Primary 37E35, 37A25}
\dedicatory{To the Anniversary of Anatole Katok, my Friend and Teacher.}

\title[Well-approximable angles and mixing]
{Well-approximable angles and mixing\linebreak[1] 
for flows on $\mathbb{T}^2$ with
nonsingular fixed points} 

\author{A. Kochergin}
\address{Department of Economics,
Lomonosov Moscow State University, Leninskie Gory,
Moscow 119992, Russia}
\email{avk@econ.msu.ru}


\date{June 14, 2004}
\revdate{August 17, 2004}

\thanks{The work was partially supported by the program ``Leading Scientific
Schools of Russian Federation", project no. NSh-457.2003.01.}

\begin{abstract}
We consider  special flows
over circle rotations with an asymmetric function having
logarithmic singularities.
If some expressions containing singularity coefficients
are different from any negative integer, then there exists a class of
well-approximable angles of rotation such that the special flow over the
rotation 
of this class is mixing.
\end{abstract}

\maketitle


Examples of smooth flows on a two-dimensional torus with a smooth
invariant measure and
nonsingular hyperbolic fixed points appear naturally in Arnold's
paper~\cite{A-E}.
The phase space of such a flow decomposes into cells bounded by closed
separatrices of regular fixed points and filled with periodic orbits,
and an ergodic component in which orbits
on one side of a fixed
point visit its neighborhood more frequently than on the other.
(See the figure, for example.)
\medskip

\begin{center}
\includegraphics[width=0.35\textwidth]{era136el-fig-1.eps}
\end{center}

\medskip
V. I. Arnold has shown that there exists a smooth closed curve transversal
to the orbits of the ergodic component. The invariant measure and 
the flow naturally induce a smooth
parameterization on the curve (this procedure was described in detail,
e.g., in \cite{K4-E}).
The first-return map is determined everywhere on the
curve, except for a finite number of points that are the points of the last
intersection of the stable separatrices with the curve.
In the induced parameterization, this map is a circle rotation. The return
time is a smooth function of the parameter everywhere except for the same
points. In the same work, it was shown that this function has logarithmic
singularities at these points since the residence time within a small
neighborhood of the nondegenerate saddle point is of the order of $\log \tau$,
where $\tau$, roughly speaking, is the distance from
the point of intersection of the orbit with
the boundary of the neighborhood
to the separatrix.

Thus, the ergodic component of such a
flow is isomorphic to a special flow $S^t$ over rotation $T$ of the circle
$\T^1=\R/\Z$
and under
a ``roof'' function with logarithmic singular points which, in general,
is asymmetric. In view of the ergodicity, the angle of rotation is
irrational. Each singularity
as usual is asymmetric since an orbit, sufficiently
close to the separatrix, visits a neighborhood of the fixed point once or
twice, depending on whether it
passes on the left or on the right
of the separatrix.
In the general case, the function is asymmetric; however, in some cases, it
may be symmetric (see definitions below).


We say that a ``roof'' function has \emph{logarithmic singularities} if
it satisfies the following conditions:

1) $f$ has $K$ singular points $\ol{x}_1,\dots,\ol{x}_K$;

2) $f\in C^1(\T^1\setminus\bigcup_{i=1}^{K}\ol{x}_i)$, $f(x)\ge c>0$;

3) for any $i=1,\dots,K$,
\begin{gather*}
f'(x)=\frac{1}{\{x-\ol{x}_i\}}(-A_i+o(1))\ \text{ for \ }x\to\ol{x}_i+0,\\
f'(x)=\frac{1}{\{\ol{x}_i-x\}}(B_i+o(1))\ \text{ for \ }x\to\ol{x}_i-0,
\end{gather*}
where $A_i$, $B_i> 0$.

Let
$$
A=\sum_{i=1}^KA_i,\quad B=\sum_{i=1}^KB_i.
$$

The function  $f$ is called
\emph{symmetric} if
$$A=B,$$
\emph{asymmetric} if
$$A\ne B,$$
and \emph{strongly asymmetric} if for any $i$
$$
\frac{A_i-B_i}{A-B}>0.
$$

For symmetric functions it is known \cite{K4-E} that
  if
  $$
  f(x)=f_0(x)+\sum_{i=1}^{K}{\left(A_i
      \log\frac{1}{\{x-\ol{x}_i\}}+ B_i
      \log\frac{1}{\{\ol{x}_i-x\}}\right)},
  $$
  where $f_0$ has a bounded variation, $A=B$, and $\rho$ admits an
  approximation by rationals with the rate $\frac{\mathrm{const}}{q^2\log q}$, 
then the special flow over the circle
  rotation by $\rho$ with ``roof'' function $f$ is not mixing.


The conjecture about the possibility of mixing in special flows
with
an asymmetric ``roof'' function was proposed
in \cite{K4-E}. Khanin and Sinai
proved it for a certain class of Diophantine
rotation
angles $\rho$.
More precisely, let
$\rho=[k_1,\dots,k_n,\dots]$~be the expansion of $\rho$ in a
continued fraction,  and $p_n/q_n$ the $n$th convergent to $\rho$.
In \cite{KhS-E}, the restriction on $\rho$ is
$$
k_{n+1}\le \mathrm{const}\ n^{1+\gamma},\quad
0<\gamma<1.
$$

In \cite{K7-E}, the mixing is proved for angles  satisfying
\begin{equation*}\tag{$*$}
  \log k_{n+1}=o\left(\log q_{n}\right). 
\end{equation*}
It is easy to show
that, if for some $\gamma>0$, perhaps
greater than 1, the
inequality $k_{n+1}\le \mathrm{const}\ n^{1+\gamma}$ holds for all $n$, 
then $\rho$ satisfies ($*$).

If $f$ is a strongly asymmetric function with logarithmic singularities
and $\rho$ is an arbitrary irrational angle, then
the special flow over
a circle rotation through the angle
$\rho$ with the ``roof'' function  $f$ is mixing
\cite{K8-E}.

In this paper we
return to the general nonsymmetric case. We
extend the theorem about mixing to some class of
well-approximable angles if the coefficients of logarithmic singularities
satisfy 
no negative integer value condition defined in Theorem~\ref{Liu} below.

One more extension of old results
is related to the definition of a strictly logarithmic
singularity which describes this singularity more precisely.

We say that the singular point $\ol x_i$ is
\emph{strictly logarithmic} if there
exists a function
$\Psi\in C^2(0,1]$
such that
$
\Psi>0, \ \Psi'>0, \ \Psi''\le 0, \ \Psi(+0)=0,
$
and in some right half-neighborhood of $\ol{x}_i$
$$
\left|f'(x)+\frac{A_i}{\{x-\ol{x}_i\}}\right|\le\Psi'(\{x-\ol{x}_i\}),
$$
and in some left half-neighborhood of $\ol{x}_i$
$$
\left|f'(x)-\frac{B_i}{\{\ol{x}_i-x\}}\right|\le\Psi'(\{\ol{x}_i-x\}).
$$

Note that, as follows from~\cite{K4-E}, if the invariant measure on
$\T^2$ has a smooth positive density, then the singularities of
a ``roof"
function of the special flow are strictly logarithmic.


Given a subset $I$ of
$\{1, 2,\ldots,K\}$ we denote
\begin{equation*}
D_I=\sum_{i\in I}\frac{B_i-A_i}{B-A}.
\end{equation*}
If $I=\varnothing$ then $D_I=0$.
Let
$I_{{\rm s}}$ be the set of numbers of strictly logarithmic singularities
and $I_{{\rm ns}}$ be the set of numbers of other singularities.

\begin{thm}\label{Liu}
Let $f$ be asymmetric and such that for any
${I}\subset \{1, 2,\ldots,K\}$ and any $l\in\N$
\begin{equation}
D_I\ne-l.
\label{-l}
\end{equation}
Assume in addition that $D_I\ne 0$ whenever
$I\cap I_{{\rm ns}}\ne\varnothing$.

Then
for any
$\rho$
satisfying
\begin{equation}
\liminf\limits_{n\to +\infty}
\frac{\log k_{n+1}}{\log q_{n}}
>\max\limits_{I:D_I<0}
\frac{\left[|D_I|\right]+1}{\left\{|D_I|\right\}},
\label{lambda}
\end{equation}
the special flow constructed over
a ${\rho}$-rotation
and under $f$ is mixing. (Here $[x]$ is the integer part of $x$, and $\{x\}$ is
its fractional part.)  
\end{thm}


\medskip

The
following theorem is an extension of the main theorem in \cite{K8-E}.

\begin{thm}\label{StAsymm-2}
If $(B_i-A_i)/(B-A)>0$ for any $i\in I_{{\rm ns}}$, and
$(B_i-A_i)/(B-A)\ge 0$ for strictly logarithmic singular points of $f$,
then for any irrational $\rho$, the special flow
constructed from
${\rho}$
and $f$ is mixing.
\end{thm}


To give a sketch of the proof we recall some notions and facts from
\cite{K8-E}.

In special flows over a circle rotation the only possible cause of mixing is
the difference in the times that various points take to get from the
``floor'' to the ``roof''. This can cause, as time passes, a small
rectangle
to be strongly stretched and almost uniformly distributed along
trajectories and hence over the phase space.

The divergence of adjacent points is described via Birkhoff sums of
the ``roof'' function
$$
f^r(x)=\sum_{k=0}^{r-1}f(T^kx).
$$
It is obvious from the relation
$S^t(x,y)=S^{y+t-f^r(x)}(T^rx,0)$,
where $(x,y)$ denotes a point of the phase space. Strong and almost uniform
distribution of a small rectangle over the phase space is ensured by
a strong and almost uniform stretching of Birkhoff sums for $r\approx t$.

Formally this is described by the theorem below.


For $x\in \T^1$,
let $\eR(t,x)$ be
the number of ``jumps" which the point $(x;0)$ has done under the action of
the special flow $S^t$
over 
the time $t$. For any measurable $X\subset\T^1$ we set
$$
\eR(t,X)=\bigcup_{x\in X}\eR(t,x).
$$



\begin{thm*}[{Sufficient condition for mixing}]\label{SufCondMix}
  Let $T$ be an ergodic circle rotation.  For some $t_0>0$ assume that
  the following objects are fixed for each $t>t_0$:
\begin{itemize}
\item[---] a finite partial partition~$\xi_t$
into closed intervals: $\xi_t=\{C\}$, where
$$
\lim_{t\to +\infty}\max_{C\in \xi_t}
\left|C\right|=0,
\quad\lim_{t\to +\infty}\mu([\xi_t])=1
$$
\emph{($[\xi_t]$ is the union of elements of $\xi_t$);}
\item[---] positive functions $\eps(t)$, $H(t)$, such that
$\eps(t)\to 0$, $H(t)\to +\infty$
as $t\to+\infty$.
\end{itemize}
If for each $t>t_0$ for any $C\in\xi_t$ and 
$r\in\eR(t,[\xi_t])\cap(t/\sqrt2,\sqrt2 t)$,
\begin{gather*}
(f^r)'(x)=M(r,C)(1+\gamma(r,x)),\\
|C||M(r,C)|\ge H(t),\ \  |\gamma(r,x)|<\eps(t),
\end{gather*}
\emph{($M(r,C)$, $\gamma(r,x)$ are real functions)},
then $S^t$ is mixing.
\end{thm*}

At first, we consider the derivatives of ``ideal logarithmic functions". Let
$u$, $v\colon\R\to\R$ be the functions with period 1 which are defined as
\begin{align*}
&u(x)=1/x \text{ if }\ x\in(0,1],\\
&v(x)=1/(1-x) \text{ if }\ x\in[0,1).
\end{align*}
Then $x_0=0$ is their singular point in $\T^1$.  One can show
that for
every $x$, except singular points of $f^r$,
\begin{align}
\begin{split}
(f^r)'(x)&=
\sum_{i\in I_{{\rm ns}}}\left(u^r(x-\ol{x}_i)(-A_i
+\alpha^-_i(r,x))+
v^r(x-\ol{x}_i)(B_i+\alpha^+_i(r,x))\right)\\
&\quad+\sum_{i\in I_{{\rm s}}}\left(-A_iu^r(x-\ol{x}_i)+(\psi_i^-)^r(x)
+B_iv^r(x-\ol{x}_i)+(\psi_i^+)^r(x) \right),
\label{f-raz}
\end{split}
\end{align}
where
$|\psi_i^-(x)|\le \Psi'(x-\ol{x}_i)$,
$|\psi_i^+(x)|\le \Psi'(\ol{x}_i-x)$,
$|\alpha^{\pm}_i(r,x)|\le \alpha(r)$, $\alpha(r)\to 0$
as $r\to +\infty$.

Suppose that
$
q_m\le r(\log q_n)^{-1/4}\quad \text{ (for }\ n>1),
\quad \sigma_n^2\log q_n\nearrow +\infty.
$$
For the set $W$, we define $U(\eps,W)=\bigcup_{x\in W}U(\eps,x)$,
where $U(\eps,x)$ is the $\eps$-neighbor\-hood of $x$.


\begin{thm}[On the ``main resonant term"]
  For sufficiently large $m$, the following
cases are possible:

\pr{1.} The ``main resonant term" is absent:
for any $s\sigma^2_m\log q_m,\quad
\sqrt{2}\sigma_m q_{m+1}\le t<\sqrt{2}q_{m+1}.
$$
Then for any $x\in V(t)$
$$
\sum_{n \ne m}Z_n^-(r,x)<16\sigma_me(r),
$$
and for $Z_m^-(r,x)$, when $r\in(t/\sqrt{2},\sqrt{2}t)$
and
$x\in {V}(t)\setminus U(\sigma_m/q_m, \pa[X^{(m)}(r)])$,
there is an alternative:
\begin{itemize}
\item[---] if $x\notin[X^{(m)}(r)]$, then $Z_m^-(r,x)<\sigma_me(r)$;
\item[---] if $x\in[X^{(m)}(r)]$, then
$
q_{m+1}\ln k_{m+1}-\sigma_me(r)<
Z_m^-(r,x)\sigma_mt\log q_m.
$
Then for any $r\in(t/\sqrt{2},\sqrt{2}t)$ and $x\in {V}(t)$
$$
\sum_{n\ne s}Z_n^-(r,x)<\sigma_me(r);
$$
for $Z_s^-(r,x)$, when $r\in(t/\sqrt{2},\sqrt{2}t)$
and
$x\in {V}(t)\setminus U(\sigma_m/q_s, \pa[X^{(s)}(r)]),$
there is an alternative:
\begin{itemize}
\item[---] if $x\notin[X^{(s)}(r)]$, then
$Z_s^-(r,x)<\sigma_me(r)$;
\item[---] if $x\in[X^{(s)}(r)]$, then
$q_{s+1}\log k_{s+1}-\sigma_me(r)<
Z_s^-(r,x)0
$$
for $C$ such that $L(C)< 0$.


Let
$\lambda>0$ be such that for   sufficiently large $m$
\begin{equation}
\frac{\ln q_{m}}{\ln k_{m+1}}\le\frac{1}{\lambda}.
\label{lam}
\end{equation}
It is easy to see that
\begin{gather*}
1<\frac{\ln q_{m+1}}{\ln k_{m+1}}\le 1+\frac{1}{\lambda}+\nu_{m+1},
\quad
\nu_{m+1}=\frac{q_{m-1}}{k_{m+1}q_m\ln k_{m+1}}.
\end{gather*}


\begin{lem}\label{D-1}
For sufficiently large $t$,
 for any
$r\in (t/\sqrt2, \sqrt2t)$, if $L(C)<0$, then
$$
\frac{e(r)}{L(C)}\in\wt{\eD}(\lambda,m)
=\bigcup_{l\in\Z_+} \bigcup_{I:D_I<0}
\left(\frac{1}{D_{I}}\Bigl(l+\frac{l+1}{\lambda}+l\nu_{m}'\Bigr),
\frac{l}{D_{I}}\right),
$$
where $\nu_{m}'=\max(\nu_{m+1}, \nu_{m})$.
\end{lem}

\begin{proof}
It will suffice to consider only cases~2~and~3 described in 
Theorem 3.

We begin with
the case 3. In this case, there exists a number
$s\sigma_mt\ln q_m.$ It is obvious that
$s=m-1$. Indeed, suppose that
$s+1q_m$, we have
$$
\frac{q_{s+1}\ln k_{s+1}}{t\ln q_{m}}<
\frac{q_{s+1}\ln k_{s+1}}{q_{m}\ln q_{m}}<
\frac{1}{k_m}\le
\frac{1}{q_m^{\lambda/(\lambda+2)}}<\frac{1}{\ln q_m}<\sigma_m,
$$
which contradicts the definition of the case 3.

Thus, in view of (\ref{DefL2})
$$
L(C)=D_{I(C)}q_{m}\ln k_{m}.
$$
From the condition
$q_{m}\ln k_{m}>\sigma_mt\ln q_m$, for sufficiently large
$m$, it follows that
$$
t<\frac{q_{m}\ln k_{m}}{\sigma_m\ln q_m}\frac{1}{D_{I}}\frac{l\ln q_m}{\ln k_{m}}\ge
\frac{e(r)}{L(C)}>
\frac{1}{D_{I}}\frac{l\ln q_m+\ln q_{m-1}}{\ln k_{m}}>
\frac{1}{D_{I}}\left(l\bigl(1+\frac{1}{\lambda}+\nu_{m}\bigr)+
\frac{1}{\lambda}\right),
$$
that is, for $r\in[lq_m, (l+1)q_m)$ \quad we have
$$
\frac{e(r)}{L(C)}\in
\left(\frac{1}{D_{I}}\bigl(l+\frac{l+1}{\lambda}+l\nu_{m}\bigr),
\frac{l}{D_{I}}\right).
$$

\medskip

The case 2 will take place if
$\sqrt{2}\max(q_m,\sigma_mq_{m+1})\le t<\sqrt{2}q_{m+1}$,
and ${r}\in\break(\max(q_m,\sigma_mq_{m+1}),{2}q_{m+1})$.

First, consider the case of $q_m<{r}\frac{1}{D_{I}}\frac{q_m\ln q_m}{q_{m+1}\ln k_{m+1}}\ge
\frac{e(r)}{L(C)}>
\frac{1}{D_{I}}\frac{\ln q_m}{\ln k_{m+1}}>
\frac{1}{D_{I}\lambda},
$$
i.e.
$$
\frac{e(r)}{L(C)}\in
\left(\frac{1}{D_{I}\lambda},0\right).
$$

Now, suppose that $q_{m+1}\le r<2q_{m+1}$ (and hence, $r\frac{e(r)}{L(C)}
>\frac{1}{D_{I}}\left(1+\frac{2}{\lambda}+\nu_{m+1}\right)
$$
at $D_I<0$, i.e.,
$$
\frac{e(r)}{L(C)}\in
\left(\frac{1}{D_{I}}\bigl(1+\frac{2}{\lambda}+\nu_{m+1}\bigr),
\frac{1}{D_{I}}\right).
$$

Combining all the cases, we obtain the conclusion of the lemma.
\end{proof}

Hence,
if
\begin{equation}
-1\notin\eD(\lambda)=\bigcup_{l\in\Z_+} \bigcup_{I:D_I<0}
\left[\frac{1}{D_{I}}\bigl(l+\frac{l+1}{\lambda}\bigr),
\frac{l}{D_{I}}\right],
\label{-1}
\end{equation}
then there exists $\theta'>0$ such that for  sufficiently large $t$,
for any $C\in\xi_t$ and  $r\in(t/\sqrt2, \sqrt2t)$
$$
\left|\frac{e(r)}{L(C)}+1\right|\ge \theta'.
$$
From this we obtain, that, if for any $I$ and any $l\in\N$,
the relation $D_I\ne -l$ is true, then there exists  $\lambda$ for which
${-1\notin {\eD}(\lambda)}$, and hence the special flow is mixing.

It is easy to 
deduce from (\ref{lambda}) the existence of $\lambda$
satisfying (\ref{lam}) and (\ref{-1}).


\bigskip

The full proof will be published in Mat. Sbornik, perhaps, in 2004.

\bibliographystyle{plain}

\begin{thebibliography}{111}
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V. I. Arnold, \emph{
 Topological and ergodic properties of closed 1-forms with
incommesurable periods},
Funkts. Anal. i Prilozh.
{\bf 25} (1991), no. 2,~1--12; English
 transl., Funct. Anal. Appl.  \textbf{25}  (1991), no. 2, 81--90.
\MR{1142204 (93e:58104)}

\bibitem{KhS-E}
K. M. Khanin and Ya. G. Sinai,
\emph{A mixing for some classes of special flows over
rotations of the circle},
Funkts. Anal. i Prilozh.
{\bf 26} (1992), no. 3, 1--21; English transl.,  Funct. Anal. Appl.  
\textbf{26} 
(1992), no. 3, 155--169.
\MR{1189019 (93j:58079)}

\bibitem{K4-E}
A. V. Kochergin, \emph{
 Nonsingular saddle points and the absence of mixing},
Math. Notes {\bf 19} (1976), 277--287.
\MR{415681 (54:3761)}

\bibitem{K7-E} A. V. Kochergin, \emph{
 Nonsingular saddle points and mixing in flows on two-dimensional
torus},
Sb. Math. {\bf 194} (2003), no. 7-8, 1195--1224.
\MR{2034533}

\bibitem{K8-E} A. V. Kochergin,
\emph{Nonsingular saddle points and mixing in flows on two-dimensional
torus II},
Mat. Sb. {\bf 195} (2004), no. 3, 15--46. (Russian)
\MR{2068956}

\end{thebibliography}
\end{document}