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\headline={\ifnum \pageno>1
\tenrm
January 26, 1999 \hfil \folio
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\define\db{dispersive billiard\ }
\define\sdb{semi-dispersive billiard\ }
\define\dbs{dispersive billiards\ }
\define\sdbs{semi-dispersive billiards\ }
\define\disp{dispersive \ }
\define\sdisp{semi-dispersive \ }
\define\bil{billiard\ }
\define\suf{sufficient\ \ }
\define\erg{ergodicity\ \ }
\define\scs{symbolic collision sequence\ \ }
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\define\sci{(\Sigma,C_i)}
\def\ahb{A_H(B)}
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\def\bp{\Bbb P}
\def\bz{{\Bbb Z}}
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\define\coK{K_{(\bar q_i,I_i,E_i)}}
\define\coKt{K_{(\bar q_i +tI_i,I_i,E_i)}}
\def\ks{\Cal S}
\def\kt{\Cal T}
\def\kv{\Cal V}
\def\ke{\Cal E}
\def\kn{\Cal N}
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\define\qm{\partial \bold {Q}^-}
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\define\pq{\partial \bold {Q}}
\define\qi{\partial \bold {Q}_i}
\define\qji{\partial \bold {Q}_{j(i)}}
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\define\mm{\partial {M}^-}
\def\g{\gamma}
\define\spix{S^{(0,\infty)}x}
\define\clc{C\ell_C (C_{ed}(t_0,P))}
\define\ctp{C_{ed}(t_0,P)}
\define\xu{U(x_0)}
\define\yu{U(y_0)}
\define\fie{\uph _{I,E}\,}
\define\pik{\pi_{2,3,4}\,}
\define\mezo{(\bold M, \Cal F, \mu)}
\define\dina{(\bold M, \Cal F, S^{\br_+}, \mu)}
\define\ddina{(\bold M, \Cal F, T^{\bz}, \mu)}
\define\kssw{\hbox{[K-S-Sz(1989)]\;}}
\define\ksst{\hbox{[K-S-Sz(1990)]\;}}
\define\kssh{\hbox{[K-S-Sz(1991)]\;}}
\define\kssn{\hbox{[K-S-Sz(1992)]\;}}
\define\sima{\hbox{[Sim(1992)-I]\;}}
\define\simb{\hbox{[Sim(1992)-II]\;}}
\define\szab{\hbox{[Sz(1995)]\;}}
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\define\sszh{\hbox{[S-Sz(1995)]\;}}
\define\sszt{\hbox{[S-Sz(1994)]\;}}
\define\ssz8{\hbox{[S-Sz(1999)]\;}}
\define\sc{\hbox{[S-Ch(1987)]\;}}
\define\bus{\hbox{[B-S(1973)]\;}}
\define\woj{\hbox{[W(1985)]\;}}
\define\blps{\hbox{[B-L-P-S(1992)]\;}}
\define\dist{\text{dist\;}}
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\define\vTm{v^{T^-}}
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\define\vTp{v^{T^+}}
\define\wtm{w^{t^-}}
\define\wtp{w^{t^+}}
\define\wTm{w^{T^-}}
\define\wTp{w^{T^+}}
\def \qf {(q_1,q_2,q_3,q_4)}
\define\sinf{S^{(-\infty,\infty)}x_0\,}
\define\xim{$x_0\in M_d$\,}
\define\mpp{M_{P_1,P_2}\,}
\define\fcf{F_{-}\cap F_{+}\,}
\define\sncx{$S^{(-\infty ,0)}_{C_i(P_1)}x_0$\,}
\define\ssabx{$\Sigma(S^{[a,b]}x)$}
\define\saby{$S^{[a,b]}y$}
\define\sabx{$S^{[a,b]}x$}
\define\spcx{$S^{(0,\infty )}_{C_i(P_2)}x_0$\,}
\define\sncy{$S^{(-\infty ,0)}_{C_i(P_1)} y$\,}
\define\spcy{$S^{(0,\infty )}_{C_i(P_2)} y$\,}
\define\steb{\left\{ S^t_{1,2}\right\}\,}
\define\sthb{\left\{ S^t_{3,4}\right\}\,}
\define\stnb{\left\{ S^t_{2,3,4}\right\}\,}
\define\ste{S^t_{1,2}\,}
\define\sth{S^t_{3,4}\,}
\define\stn{S^t_{2,3,4}\,}
\define\pikv{\pi^V_{2,3,4}\,}
\define\sni{S^{(-\infty,0)}\,}
\define\snis{S^{(-\infty,0)}_*\,}
\define\snin{S^{(-\infty,0)}_{2,3,4}\,}
\define\spi{S^{(0,\infty)}\,}
\define\spis{S^{(0,\infty)}_*\,}
\define\spie{S^{(0,\infty)}_{1,2}\,}
\define\spih{S^{(0,\infty)}_{3,4}\,}
\define\spin{S^{(0,\infty)}_{1,3,4}\,}
\define\vk{(v_1,v_2,v_3,v_4)\,}
\define\qvi{(q_1,v_1,I-v_1)\,}
\define\qvk{(q_1,q_2,v_1,v_2)\,}
\define\qvio{(q_1^0,v_1,I-v_1)\,}
\define\qviz{(q_1^0, v_1^0, I-v_1^0)\,}
\define\traj{S^{[0,T]}x_0}
\define\flow{\left(\bold{M},\{S^t\},\mu\right)}
\define\proj{\Bbb P^{\nu-1}(\Bbb R)}
\define\ter{\Bbb S^{d-1}\times\left[\Bbb P^{\nu-1}(\Bbb R)\right]^m}
\define\sphere{\Bbb S^{d-1}}
\define\pont{(V;h_1,h_2,\dots ,h_m)}
\define\projm{\left[\Bbb P^{\nu-1}(\Bbb R)\right]^m}
\define\pontg{(V_0;g_1,g_2,\dots ,g_m)}
\define\szorzat{\prod\Sb i=1\endSb\Sp m\endSp \Cal P_i}
\define\szorzatk{\prod\Sb l=1\endSb\Sp k\endSp \Cal P_{i_l}}
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\document
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%\advance\baselineskip 8pt
\noindent
January 26, 1998 \par \noindent
\bigskip \bigskip
\centerline{\bf Ball-Avoiding Theorems}
\bigskip \bigskip \bigskip
\hbox{}
\centerline {{\bf Domokos Sz\'asz}
\footnote{Research supported by the Hungarian National Foundation for
Scientific Research, grants OTKA-16425 and OTKA-26176.}}
\centerline{Mathematical Institute of the Hungarian Academy of Sciences}
\centerline{H-1364, Budapest, P. O. B. 127, Hungary}
\centerline{E-mail: szasz\@math-inst.hu}
\bigskip \bigskip
\hbox{\centerline{\vbox{\hsize 8cm {\bf Abstract.}
Consider a nice
hyperbolic dynamical system (singularities not excluded). Statements
about the topological smallness of the subset of orbits,
which avoid an open subset of the phase space
--- for every moment of time, or just for a not too
small subset of times --- play a key role in showing hyperbolicity or
ergodicity of semi-dispersive billiards, in particular, of hard ball systems.
Beside surveying the characteristic results, called ball-avoiding theorems,
and giving an idea of
the methods of their proofs, their applications are also illustrated. Further
we also discuss analogous questions (which had arisen, for intance, in number
theory), when Hausdorff dimension is taken
instead of the topological one. The answers strongly
depend on the notion of dimension which is used.
Finally, ball avoiding subsets are naturally
related to
repellers extensively studied by physicists. For the interested reader
we also sketch
some analytic and rigorous results about repellers and escape times.}}}
\newpage
\bigskip \bigskip
\heading
{\bf CONTENTS}
\endheading
\noindent
1. Introduction.
\smallskip
\par {\bf I. Weak and Strong Ball-Avoiding Theorems}
\smallskip
\par \noindent 2. An (Abstract) Weak Ball-Avoiding Lemma
\par \noindent 3. Facts from Topological Dimension Theory
\par \noindent 4. A Theorem of Smale and Williams for Anosov Diffeomorphisms
\par \noindent 5. A Strong Ball-Avoiding Theorem for Hyperbolic Systems
(with or without Singularities)
\smallskip
\par {\bf II. Ball-Avoiding Theorems and Hyperbolic Properties}
\smallskip
\par \noindent 6. Hyperbolic Properties of Hard Ball Systems
\roster
\item "{---}" Isometry to Semi-Dispersive Billiards
\item"{---}" Local Ergodicity of Semi-Dispersive Billiards
\item "{---}" Richness of a Symbolic Collision Sequence
\item"{---}" The Role of Ball-Avoiding Theorems in Proving
Hyperbolicity or Ergodicity
\endroster
\par \noindent 7. Interlude: An Instructive Example
\par \noindent 8. Results for Hard Ball Systems
\roster
\item "{A.}" Dynamical Method
\item "{B.}" Mechanical Method
\endroster
\par \noindent 9. Hyperbolic Properties of Cylindric Billiards
\par
\smallskip
{\bf III. Related Directions}
\smallskip
\noindent 10. Replacing Topological Dimension with the Hausdorff-one
\par \noindent 11. Ball-Avoiding in Physics: Open Systems
and Repellers
\par \noindent
\heading
1. Introduction.
\endheading
\bigskip \bigskip
The seminal work of Chernov and Sinai \sc
not only established the K-property
of \dbs in the general, multidimensional case,
but --- through their Theorem on Local
Ergodicity for \sdbs --- also opened the possibility
to think about showing the
K-property of \sdbs. Indeed, by using this fundamental tool,
in 1989, A. Kr\'amli, N. Sim\'anyi and the present author \kssw
could first show the K-property of a \bil, which was \sdisp but not \disp.
Our method, which has later been further developed in a series of works
(for a survey of the results see [Sz(1996)]),
consists of three essential parts using dynamical--topological,
geometric--algebraic, and finally dynamical--measure-theoretic tools,
respectively.
The dynamical--topological methods of these proofs are distilled
in so-called
{\it ball-avoiding theorems}, whose content we are just going to
formulate, and which will be the content of this survey.
Assume $\dina$ is a semigroup of endomorphisms (or $(\bom, \Cal F, S^\br, \mu)$
is a group of automorphisms) of a probability space
$\mezo$. For formulating topological statements, we will, in
general, assume that $\bom$ is a Riemannian manifold with or without
boundary. Most of our methods will use some hyperbolicity and/or
mixing properties of the dynamics involved.
Fix an arbitrary subset $H$ of $\br_+$ (or of $\br$)
and a subset $B \subset \bold M$.
For $B$ and $H$ given in this way, the ball-avoiding subset $A_H(B)
\subset \bold M$ is defined as follows:
$$
A_H(B)= \{x \in \bold M:\ S^Hx \cap B =\emptyset\}.
$$
In words, it consists of phase points whose orbits avoid the subset $B$
in prescribed moments of time ($B$, in general, need not be a ball, but
often it is, and
the term ball-avoiding already has traditions). If $B$ is not too
small, e. g. it is open, then $\ahb $, as a collection of nontypical
trajectories, is expected to be small.
{\it Ball-avoiding
theorems}
claim that, by assuming that $B$ is not very small and $H$ is
unbounded (or semi-unbounded, at least),
$A_H(B)$ is small in a well-defined sense (i. e. its topological
codimension is at least one or two, and, moreover,
$\mu\{A_H(B)\}=0$) ---
under weaker or stronger assumptions on the hyperbolic and/or ergodic
behaviour of the dynamics. It is worth stressing that, though some
general results have only been formulated for semi-dispersive billiards,
their validity is wider: they are true for a class of
`hyperbolic' systems with singularities possessing a smooth invariant
probability measure.
If $\bom$ is a separable and metrizable space, then let
$\{B_i:\ i=1, 2, \dots\}$ be a basis of the topology in $\bom $ (then each
$A_H(B_i)$ is closed provided that the group $S^\br$ is continuous). Denote
$$
ND:=\ \{x \in \bom:\ S^{\br}x\ \text {is not everywhere dense
in} \ \bom\} \tag 1.1
$$
Plainly, $ND= \cup_{i} (A_{\br_-}(B_i)\cap A_{\br_+}(B_i))$.
If one shows that each
$A_{\br_-}(B_i)\cap A_{\br_+}(B_i)$ is a
zero-measure subset of codimension two, then
$ND$ will necessarily be {\it slim} (for the definition see section 3),
i. e. topologically small.
Though the question, ball-avoiding theorems answer, is natural, in
this form they seem to have not been treated before \kssw. In the particular
case $H = \br$ \ the set $A_\br(B)$ is an invariant subset. These sets were
used by Smale (cf. [H(1970)]) and later by others to analyze possible
dimensions of
compact, proper invariant subsets of a hyperbolic diffeomorphism.
The difference between their treatment of the problem and between ours
reflects the very difference between smooth (Anosov) systems and those
with singularities.
On the other hand, there is
a very active and interesting direction of research
investigating, in
particular, the same subsets $ND$ from a different point of view.
These results generalize a classical theorem of Jarnik, [J(1929)] and of
Besicovitch, [B(1934)]
claiming that the set of badly approximable (or Diophantine) numbers
in the interval $[0, 1]$ has Hausdorff dimension 1. The typical result
then claims that the {\it Hausdorff-dimension} of the
subset $ND$ is maximal, i. e. agrees with $\dim \bom$. In other
words, despite the fact that these orbits are non-typical,
nevertheless, the Hausdorff-dimension does not sense this atypicality.
Also, in the last
years
physicists got interested in open systems, e. g. in open billiards,
which
actually live on a ball-avoiding subset of the phase space of a closed
billiard. As a consequence, these systems got
also investigated from the mathematical point of view. Since the interest
of their authors was different from ours (cf. [Ch-M-T(1998)]) we will be
satisfied to give a brief account of their main characteristic results.
This work is partitioned into three parts. In the first one, consisting of
sections 2-5, the simplest ball-avoiding theorems are presented: a weak
one in section 2, and --- after a brief summary of some useful notions
from topological dimension theory given in section 3 --- some strong ones in
sections 4 and 5. In the second part, consisting of sections 6 to 9, first the
relevance of
ball-avoiding theorems for hard ball systems is explained. Then various
forms of them are surveyed. Our additional aim is to present
the different methods used in their proofs, or at least to hint to them,
and to also collect the most interesting open problems.
Finally, in Part III some related directions mentioned above are reviewed.
\bigskip \bigskip
\heading
{\bf I. Weak and Strong Ball-Avoiding Theorems}
\endheading
\bigskip \bigskip
\heading
2. An (Abstract) Weak Ball-Avoiding Lemma
\endheading
\bigskip \bigskip
Let $\dina$ be a semigroup of endomorphisms of a probability space
$\mezo$.
Fix an arbitrary subset $H$ of $\br_+$ satisfying $\sup H=+\infty$.
\proclaim{Lemma 2.1 \kssw} If the semigroup $\Cal S^{\br_+}$ is mixing,
then, for any $B \in \Cal F$ with $\mu\{B\}>0$, one has
$$
\mu\{A_H(B)\}=0.
$$
\endproclaim
Since the proof is extremely simple, it will be presented below.
\demo {Proof} Denote
$$
A^\tau_H(B):= \{x \in \bold M:\ S^{H \cap [0,\tau]}x \cap B
=\emptyset\}.
$$
Then, on one hand,
$$
\mu \{A^\tau_H(B)\} \searrow \mu \{A_H(B)\} \tag 2.2
$$
if $\tau \to \infty$. On the other hand, for every $t \in H$, we have
$$
\mu \{A^\tau_H(B) \cap \{S^tx \notin B\}\} \ge \mu \{A_H(B)\} \tag 2.3
$$
Then, by mixing and (2.2), (2.3) leads to
$$
\lim_{\tau \to \infty} \lim_{t \to \infty, t \in H}
\mu \{A^\tau_H(B) \cap \{S^tx \notin B\}\} =
\mu \{A_H(B)\} \mu \{B^c\} \ge \mu \{A_H(B)\}
$$
implying $\mu \{A_H(B)\} = 0$ for $ \mu\{B\}>0.$
\qed \enddemo
\medskip
\noindent {\bf Remark.} Any irrational rotation of $\br/\bz$ serves as an
example of an ergodic automorphism for which the claim of the Lemma is
not valid. Different is the situation if $H=\br$ since then ergodicity
is, of course, sufficient to imply $\mu \{A_H(B)\} = 0$.
\medskip
\noindent {\bf Remark.} The proof of the
Lemma immediately implies that its analogue for
discrete time semigroups $T^{\bz_+}$ is also true.
\bigskip \bigskip
\heading
3. Simple Facts from Topological Dimension Theory
\endheading
\bigskip \bigskip
Here we briefly summarize some necessary notions and facts from
topological dimension theory (for details see [E(1978)] or [H-W(1941)]).
Assume first, in general, that $X$ is a separable metric space. We will
denote $\dim
X$ the small inductive topological dimension of $X$ whose recursive
definition will just be recovered.
\proclaim {Definition 3.1}
\roster
\item "{(i)}" $\dim X= -1$ if and only if $X= \emptyset$;
\item "{(ii)}" $\dim X \le n$ if and only if there exists a basis
$\Cal U$ of
open neighbourhoods for $X$ such that for every $U \in \Cal U$
one has $\dim \partial U \le n-1 \ \ (n =0, 1, 2, \dots )$;
\item "{(iii)}" $\dim X=n$ if and only if $\dim X\le n$ and it is not
true that $\dim X \le n-1$.
\endroster
\endproclaim
\proclaim {Definition 3.2} If $A \subset X$, and for some natural number
$k$ one has $\dim A \le \dim X -k$, then we say that the topological
codimension of $A$ in $X$ is at least $k$ (or often we briefly say
that the topological codimension is $k$). \endproclaim
>From now on we assume that $\bom$ is a connected, smooth manifold (boundary
permitted) and $\mu$ is a smooth measure on $\bom$.
\proclaim {Proposition 3.3} For any $A \subset \bom$, $\dim A \le \dim \bom -1$
(in other words, the topological codimension of $A$ in $\bom$ is at least
1) if and only if $Int A =\emptyset$.
\endproclaim
\proclaim {Proposition 3.4} If $F \subset \bom$ is closed, then the
following statements are equivalent:
\roster
\item "{(i)}" $\text {\rm codim}_{\bom}\ F \ge 2$;
\item "{(ii)}" $F \neq \bom$ and, for every open connected set $G
\subset \bom$, the difference set $G \setminus F$ is also connected;
\item "{(iii)}" $\text {\rm Int } F = \emptyset$ and for
every point $x \in \bom$ and for any neighborhood
$V$ of $x$ in $\bom$
there exists a smaller neighborhood $W \subset V$ of the point
$x$ such that, for every
pair of points $y, z \in W \setminus F$, there is a continuous curve
$\gamma$
in the set $V \setminus F$ connecting the points $y$ and $z$.
\endroster
\endproclaim
For the main applications of strong ball-avoiding theorems we need
another concept of topological smallness closely related to
being of codimension two (this
will be clear from the content of section 5).
\proclaim
{Definition 3.5 \kssw}\ We say that $A \subset \bom $ is a {\sl slim}
subset
if and only if it is the subset of an $F_\sigma$
zero-set of codimension at least two ($A$ is a zero-set if
$\mu\{A\}=0$). \endproclaim
By their definition, slim subsets of $\bom$ form a
$\sigma$-ideal. The key property of slim subsets is expressed by the
following
\proclaim {Proposition 3.6 \kssw} \ If $\bom$ is connected, and $A$ is
slim, then $\bom \setminus A$ contains an arcwise connected
$G_\delta$-set of full measure.
\endproclaim
In applications, in particular in the inductive arguments, the following
integrability property of codimension two subsets is often very useful.
\proclaim {Proposition 3.7 \ \ \kssw}
If $\bom = \bon_1 \times \bon_2$, where $\bon_1$ and $\bon_2$ are connected
smooth manifolds, and $F
\subset \bom$\ is a closed subset such that, for every $w\in
\bon_1$, the (closed) section $F_w:=\ \{p \in \bon_2:\ (w,p) \in F
\}$ obeys
$$
\text {\rm codim}_{\bon_2}\ F_w \ge 2,
$$
then
$$
\text {\rm codim}_\bom\ F \ge 2.
$$
\endproclaim
\bigskip \bigskip
\heading
4. The Smale-Williams Theorem for
Anosov-diffeomorphisms
\endheading
Assume $\bom$ is a smooth Riemannian manifold and $T: \bom \to \bom$ is
an Anosov $C^{1}$-diffeomorphism.
Smale and Williams (see [H(1970)]) proved the following nice theorem.
\proclaim {Theorem 4.1} Assume that the set of periodic points of $T$ is
dense in $\bom$. If $F$ is a compact invariant subset of $\bom$
satisfying $\text {\rm codim}_\bom\ F \ge 1$, then $\text {\rm
codim}_\bom\
F \ge 2 $.
\endproclaim
The combination of Theorem 4.1 with our weak Lemma 2.1 provides a strong
ball-avoiding statement for smooth systems:
\proclaim {Corollary} For any $B \neq \emptyset \subset
\bom$ open, $A_\Bbb Z(B)$ is a closed set of topological codimension is at
least two.
\endproclaim
\demo {Proof of the Corollary} Topological transitivity, the
invariance of $A_\Bbb Z (B)$, and the openness of $B$ imply that
$\text {\rm Int} A_\Bbb Z (B) \neq \emptyset$. Proposition 3.3 then proves the
claim.
\enddemo \qed
\demo {Proof of Theorem 4.1} Throughout the whole paper we will denote by
$\{\gamma^s\}$ and $\{\gamma^u\}$ the invariant foliations defined by the
dynamics in question, and by
$\gamma^s_{\varepsilon}(x)$ and $
\gamma^u_{\varepsilon}(x)$ the local invariant manifolds of size $\varepsilon$
through the point $x$.
Denote by $\Cal P$ the set of periodic
points of $\bom \setminus F$. We use the following simple statements:
\noindent {\it Claim 1.} $\Cal P$ is dense in $\bom$.
\noindent {\it Claim 2.} If $x \in \Cal P$, then $\gamma^u(x) \cap F =
\gamma^s(x) \cap F = \emptyset$.
These claims easily provide the truth of the Theorem. Indeed, let $y \in
F$ and choose $\varepsilon > 0$ small. The foliations
$\{\gamma^u\}$, $\{\gamma^s\}$
define a local product structure and using it we can consider a
parallelogram $\gamma^u_\varepsilon (y) \times \gamma^s_\varepsilon (y)$.
Moreover,
we define $ F_0 = F \cap
(\gamma^u_\varepsilon(y) \times \gamma^s_\varepsilon (y))$.
By Claim 2,
for any $x \in \Cal P$, $(\gamma^u(x) \cup \gamma^s(x)) \cap F = \emptyset$,
and,
consequently, $F_0 \subset
(\gamma^u_\varepsilon (y) \setminus \cup_{x \in \Cal P}\gamma^s(x))
\times
(\gamma^s_\varepsilon (y) \setminus \cup_{x \in \Cal P}\gamma^u(x))$.
Claim 1 and Proposition 3.3 then say that the factors of the previous product
set have codimension at least one each. Hence the Theorem follows by the
product theorem (cf. Theorem III.4 of [H-W(1941)]).
Let us now prove Claim 1. Take an arbitrary open subset $G$ of $\bom$. The open
set $G \setminus F$ is not empty for otherwise we would have $\dim F = n$.
Since $\Cal P$ was dense in $\bom \setminus F$, we also have
$(G \setminus F) \cap \Cal P \neq \emptyset$.
Turn next to Claim 2. We prove $\gamma^s(x)
\cap F = \emptyset$ for an arbitrary $x \in \Cal P$. Assume
$T^px =x$. Select an open neighbourhood $G$ of $x$ disjoint from $F$. By
invariance, $(\cup_{n \in \Bbb Z} T^n G) \cap F = \emptyset$. Now for any $y
\in \gamma^s(x)$\ \
$\rho(T^{kp}y, x) \to 0$ if $k \to \infty$, and thus, for $k$
sufficiently large, $T^{kp}y \in G$ implying $y \notin F
$. \enddemo \qed
\noindent {\bf Remark 4.2.} After the aforementioned
result, the study of compact invariant
subsets was continued --- among others by Franks [F(1977)], Hancock
[H(1978)] and
Ma\~n\'e [M(1978)]. Since the sets $A_\br(B)$ provide natural examples of
compact, invariant subsets --- in fact, all compact invariant subsets are
of this form ---, this description has been
used by several authors. In particular, for every $0 \le k \le d-2$,
Przytycki [P(1980)] found examples of
sets $B_k$ such that $\dim A_\br(B_k) = k$.
\heading
5. A Strong Ball-Avoiding Theorem for Hyperbolic Systems
(With or Without Singularities)
\endheading
\bigskip \bigskip
For simplicity, we formulate the theorem for discrete time groups
$\ddina$
of hyperbolic systems since the generalization to continuous time
is straightforward. Our setup is that $\bom$ is a compact
$C^\infty$-manifold and $\mu$ is a smooth, invariant probability measure.
We also want to permit singularities as it is done in [L-W(1995)] or
in [Y(1998)] or in [Ch(1998)]. For saving space, we do not list the conditions
formulated in these works since we only use some standard consequences of them.
Namely: the $\mu$-a. e. existence of the local invariant manifolds and the
absolute continuity of the canonical isomorphism between them, and further
the simple fact that if in case of singularities we define trajectory branches
as it is described, for instance, in ????, then can always be considered
continuous on these trajectory branches.
On the other hand, the kind of hyperbolicity needed
will be implicitly ensured by our assumptions. Start with the
corresponding definition.
\proclaim {Definition 5.1} A point $x \in \bom$ is called {\sl a zigzag
point} if one can find arbitrary small open neighbourhoods $U$ of $x$
such that for every zero-set $A \subset \bom$ there exists
another zero-set $A'\supset A$ with the property: for every $y, y' \in
U \setminus A'$ there exists a chain (also called Hopf-chain)
$$
\g_{\text {loc}}^u(z_0), \g_{\text {loc}}^s(z_1), \g_{\text {loc}}^u(z_1),
\g_{\text {loc}}^s(z_2), \dots, \g_{\text {loc}}^u(z_{n-1}),
\g_{\text {loc}}^s(z_n)
$$
(here $z_0=y, z_n=y'$) of local unstable and stable invariant manifolds
inside $U$ such that each intersection
$$
\g_{\text {loc}}^u(z_i)\cap
\g_{\text {loc}}^s(z_{i+1}) \qquad \qquad \qquad (i=0, \dots, n-1)
$$
and
$$
\g_{\text {loc}}^s(z_i)\cap \g_{\text {loc}}^u(z_i) =\{z_i\}
\qquad \qquad \qquad (i=1, \dots, n-1)
$$
consists of exactly one point belonging to $U\setminus A'$.
\endproclaim
The following theorem generalizes Lemma 4.3 of \kssw, and its proof is also
based on its ideas.
\proclaim {Theorem 5.2} Assume that
\roster
\item "{(i)}" the group $\ddina$ is mixing;
\item "{(ii)}" for the subset $Z$ of zigzag points of $\bom$, $\bom
\setminus Z$ is slim;
\item "{(iii)}" $B\neq \emptyset\ (\subset \bom)$ is open;
\item "{(iv)}" $H\ (\subset \bz)$ satisfies\ $ \sup H = -\inf H =
\infty.$
\endroster
Then $A_H(B)\ (\subset \bom)$ is a closed zero-set of codimension at
least two.
\endproclaim
\noindent {\bf Remark 5.3.} For the first glance,
condition (ii) of Theorem 5.2, as formulated, might seem too restrictive,
but, fortunately, this is not the case.
In the case of hyperbolic systems with singularities, with billiards included
(cf. [L-W(1995)]),
the singularities, to be denoted by $\Cal S$
(in other words the set of points where $T$ or $T^{-1}$ is not
smooth)
form one-codimensional submanifolds of the phase space. Let us denote
$$
\Delta_n:= \cup_{-n \le k 0\ & \text
{\ and\ } \inf \{n\in H:\ T^nx\in \tilde B\}=-\infty \\ & \text
{\ and\ } \sup \{n\in H:\ T^nx\in \tilde B\}=\infty \}.
\endaligned
$$
Here $\rho^{u,s}(x)$ denotes the inner radius of the local unstable
(stable) invariant manifold $\g^u(x)$\ ($\g^s(x)$) through $x$. By (ii)
and Lemma 2.1 (this presupposes (i)) we have $\mu\{D\}=1$.
3. Since, by (ii), non-zigzag points make a slim subset, by
Lindel\"of's theorem, it
is sufficient to check that every zigzag point $z$ has a neighbourhood
$U=U(z)$ such that $A_H(B)\cap U$ is slim. This is what we do. Fix thus
$z$ and its neighbourhood according to Definition 5.1 in such a way that
$\text {diam}\ U < r/2.$ To $A=U\setminus D$ select $\tilde A\supset A$
according to the same definition. We claim that every pair of points:
$y, y' \in U\setminus \tilde A$ can be connected by a curve belonging
to $U\setminus A_H(B)$. Since $A_H(B)$ is closed, both $y$ and $y'$ have
neighbourhoods in $U$ disjoint of $\ahb$. Also, since $U\setminus \tilde
A$ is dense in $U$, we can choose $\tilde y$ and $\tilde y'\ (\in
U\setminus \tilde A)$ in these neighbourhoods and connect $y$ with
$\tilde y$ and analogously $y'$ with $\tilde y'$ inside these tiny
neighbourhoods not intersecting $\ahb$.
Connect now $\tilde y$ and $\tilde y'$ with a Hopf-chain ensured by
Definition 5.1. Since $\text {diam}\ U < r/2$, we know that the outer
diameters of all local manifolds figuring in the chain are $< r/2$.
Observe that the property that the intersection points $w$ belong to
$U\setminus \tilde A$ ensures that they belong to $D$. This implies that for
infinitely many $n\in H \cap \bz_+$\ one has $T^nw\in \tilde B$. Then
for $n$ large enough, $T^n\g^s(w)\subset B$ holds, too, implying that
$\g^s(w)\cap \ahb=\emptyset$. Analogously, for the unstable local
manifolds figuring in the chain we have $\g^u(w)\cap \ahb=\emptyset$ and
thus the desired connection between $\tilde y$ and $\tilde y'$ is, indeed,
constructed. \qed \enddemo
\noindent {\bf Remark.} Compare Theorem 5.2 with the Corollary of the
Smale-Williams Theorem 4.1. Instead of requiring the density of periodic
points we have a smooth, invariant and
mixing measure. Furthermore, we also permit singular systems, and our
assumption on $H$ is much weaker, for it can even have zero density.
An immediate consequence is the following
\proclaim {Corollary} Assume that $(\bom, \Cal F, S^\br, \mu)$ is a
group of automorphisms satisfying the conditions of Theorem
5.2 suitably modified to the continuous time case (in particular, $H \subset
\br$, and otherwise satisfies the same assumtions). Then $A_H(B)$ is a
closed zero-set of codimension at least two.
\endproclaim
\bigskip
By copying this proof one can give a simple generalization of Theorem
5.2, which will be applied in section 6. Namely, let $B_-, B_+
\ (\subset \bom)$, and define
$$
A_H(B_-, B_+):= \{x\in \bom:\ S^{H_-}x\cap B_-= S^{H_+}x\cap B_+
=\emptyset\}
$$
where $H_- := H \cap \br_-$ and $H_+ := H \cap \br_+$
\proclaim {Theorem 5.4} Assume that beside (i), (ii) and (iv) of Theorem
5.2, the following condition is satisfied
(iii)$^*$ $B_-\neq \emptyset$ and $B_+\neq \emptyset
\ (\subset \bom)$ are open;
Then $A_H({B_-, B_+})$ is a
closed zero-set of codimension at least two.
\endproclaim
\demo {Proof} The same as that of Theorem 5.2 with the natural
modification that now we select $\tilde B_-\subset B_-$ and $\tilde
B_+\subset B_+$ in such a way that $d(\tilde B_-, B_-) \ge r/2$ and
$d(\tilde B_+, B_+) \ge r/2$, and define
$$
\aligned
D:= \{x\in \bom:\ \rho^{u,s}(x)>0\ & \text
{and} \inf \{n\in H:\ T^nx\in \tilde B_-\}=-\infty \\ & \text
{and} \sup \{n\in H:\ T^nx\in \tilde B_+\}=\infty \}.
\endaligned
$$
\qed \enddemo
\bigskip \bigskip
\heading
{\bf II. Ball-Avoiding Theorems and Hyperbolic Properties}
\endheading
\bigskip \bigskip
\heading
6. Hyperbolic and Ergodic Properties of Hard-Ball Systems
\endheading
\bigskip \bigskip
\heading
Isomorphy to semi-dispersive billiards
\endheading
The main aim of this section to provide a motivation and explanation how
ball-avoiding
theorems enter in proofs of hyperbolicity and ergodicity of hard-ball
systems, or more generally, of \sdbs. Consequently, in our exposition
the details are surrendered to this goal.
Let us assume, in general, that a system of $N(\geq 2)$ balls
of unit mass and of radii
$r>0$ are given on ${\t}^\nu$, the $\nu$-dimensional unit torus
$(\nu\geq 2)$. (The assumption that the masses and the radii are
identical is not an essential restriction for our purposes.) Denote the
phase point of the $i$'th ball by
$(q_i,v_i)\in {\t}^\nu\times{\br}^\nu$. The configuration space
$\tilde
{\q}$ of the $N$ balls is a subset of ${\t}^{N\cdot\nu}$: from
${\t}^{N\cdot\nu}$ we cut out $\left({N\atop 2}\right)$ cylindric
scatterers
$$
\tilde C_{i,j} =\left\{ Q=(q_1,\dots,q_N)\in {\t}^{N\cdot\nu}:\Vert
q_i-q_j\Vert <2r\right\}, \tag 6.1
$$
\noindent $1\leq i0$ such that
$\Sigma(S^{(0, t)}x)=\Sigma$ and, moreover,
$S^{t+\br_+}x$ is partitioned by $P^+$. Encouraged by the success of the
previous argument, one is inclined to hope that the method of pasting also
permits to settle
\proclaim {Conjecture 8.4} For any $N \ge 3, \nu \ge 2$, for an arbitrary \scs
$\Sigma$ and for any pair $P^-, P^+$ of non-trivial two-class partitions,
$F(P^-, \Sigma, P^+)$ is a closed zero-set of codimension two.
\endproclaim
\subheading {Remark 8.5.a} The statement of Conjecture 8.4 immediately implies
that for any fixed $C$, $\Pi_C$ is a slim subset (for the definition of
$\Pi_C$ see (6.11)), which is exactly assumption (2) of Theorem 6.17.
Though I strongly believe that the conjecture is true, nevertheless,
the method of pasting in its present form is not strong enough
to prove Conjecture
8.4. The reason is, roughly speaking, that one still can consider the
unstable manifolds for the subdynamics restricted to the classes of $P^-$
in the time interval $(-\infty, 0]$ and the stable manifolds for the
subdynamics restricted to the classes of $P^+$ in the time interval
$[t, \infty)$. It is, however, hard to see why the absolute continuity
and transversality
statements necessary to formulate the zigzag properties, so basic to repeat
the idea of Theorem 5.4, would hold.
\subheading {Remark 8.5.b} In \kssh, the statement of Theorem 8.1
is also settled for the case $N=3, \nu=2$. However, for obtaining
it for this particular case, one also had to verify
Conjecture 8.4 for the case of a one-element \scs $\Sigma=(\sigma)$. This
was actually done in \kssh through a concrete analysis of the concrete
situation and so far it is not clear how this argument generalizes.
Having seen the limitations of the method of pasting, we will now turn to
another method, which we call the mechanical method.
\bigskip
\centerline {\it B. Mechanical Method}
\bigskip
\bigskip
The mechanical method was elaborated by Sim\'anyi in \sima.
It will be presented in the simple case of a weak type theorem
borrowed from \ssz8. A novelty and an essential
advantage of the upcoming formulation is that it is
absolute, i. e. it is not inductive; afterwards we will also see
inductive statements. A non-inductive formulation is needed
if one is only able to show hyperbolicity
of hard ball systems since this is a weak property
to permit a possible induction. For the convenience of the reader and the
brevity of exposition, the setup is simplified to the case when the
masses of the balls are identical, though, in essence, the assumption on
the identity of masses is only a minor technical one.
Denote by $R^*=R^*(N, \nu)$ the maximal number $R$ such that for every
$r\in
(0, R)$ the interior of the
configuration domain $\boq$ of the hard ball system is
connected.
\proclaim{Theorem 8.6 \ssz8}
Consider a system of $N$ $(\ge 3)$ particles on the $\nu$-torus
$\bt^\nu$ ($\nu\ge 2$) satisfying $r < R^*$. Let $P=\{P_1,P_2\}$ be
a
given, two--class partition of the $N$ particles, where, for simplicity,
$P_1=\{1, \dots, n\}$ and $P_2=\{n+1, \dots, N\}$\ \ $(n 0$
$$
\liminf _{R\to \infty} \inf _H \inf _{z \in H}
\dfrac{\text {\rm meas}\left(B_R(z) \cap \Cal L_{2r}\right)}
{\text {\rm meas}\left(B_R(z)\right)} \ge \gamma(\vec n). \tag 8.13
$$
\endproclaim
\medskip
Assume that the statement (8.11) is not true, i. e. the measure of the
subset $K$ of $ S^{\nu-1}$ described by (8.11) is positive.
Select then and fix a Lebesgue
density point $\vec n$ of $K$ with the property that at least one ratio
of the components of $\vec n$ is irrational. Denote by
$G_\varepsilon \subset S^{\nu-1}$ the ball of radius $\varepsilon$
around $\vec n$. By (8.13) we can choose $R_0$ so large that for $R\ge
R_0$ $$
\inf _H \inf _{z \in H}
\dfrac{\text {\rm meas}\left(B_R(z) \cap\Cal L_{2r}\right)}
{\text {\rm meas}\left(B_R(z) \right)} \ge \frac{\gamma(\vec n)}{2}.
$$
The set $\lambda tG_\varepsilon$ can be arbitrarily well approximated by a
ball of radius
$\lambda t\varepsilon=R$ ($R$ is fixed, $R\ge R_0$) in the hyperplane
orthogonal to $\vec n$ through the point
$\lambda t\vec n +f(t)$ if only $t$ is sufficiently large.
Consequently, if $ R \ge R_0$, then by choosing $t$ sufficiently large
and at the same time putting $\varepsilon = (\lambda t)^{-1}R$, we have
$$
\dfrac{\text {\rm meas}\left((\lambda tG_\varepsilon+f(t))\cap
\Cal L_{2r}\right)}{\text {\rm meas}\left((\lambda tG_\varepsilon+f(t)
\right)} \ge \frac{\gamma(\vec n)}{4}.
$$
But this inequality contradicts to the fact that $\vec n$ was chosen as
a Lebesgue density point of the subset $K\subset S^{\nu-1}$.
Hence Theorem 8.6 follows. \qed
\bigskip
\subheading {Remark 8.14} An essential advantage of the formulation, and
of the mechanical method as well, is
that the definition of $F_+$ only uses the ball avoiding property of the
non-negative semi-trajectory. For a weak theorem this is not surprising
(cf. Lemma 2.1) but for strong theorems this is a great advantage over
results like Theorem 5.2. Actually, Sim\'anyi used the mechanical
method to show
\proclaim{Theorem 8.15 \sima} Let $(M,\{S^t\},\mu)$ be the {\it standard
hard ball
flow} of $N$ $(\ge 3)$ particles on the unit torus $\bt^\nu$ $(\nu\ge 2)$.
Suppose that $r < R^*$.
Assume that for all $n0$ and with the property
that for every $i=1,\dots ,n$ either $L_i\subset B_1$ or
$L_i\subset B_2$.
We say that the symbolic collision sequence
$\Sigma=(\sigma_1, \dots, \sigma_n)$ is {\sl C-rich}, with $C$ being
a natural number, if it can be decomposed into at least $C$ consecutive,
disjoint collision subsequences in such a way that each of them
is connected.
\endproclaim
\noindent {\bf Remark 9.6.} The condition of connectedness is exactly
identical to the Orthogonal Non-Splitting Property, formulated in
[S-Sz(1998)], of the system of
subspaces $L_1, \dots, L_n$. Moreover,
by Theorem 4.6 and Proposition 4.9 of the same work, in the particular case
of hard ball systems our Definition 9.5 reduces precisely to Definition 6.5
given above.
\proclaim {Conjecture 9.7} For an arbitrary natural number $C$, the subset
of orbits whose \scs is not $C$-rich, is a slim
subset of $\bom$.
\endproclaim
Finally we formulate a stronger conjecture than the previous one. In principle
it is adapted to a possible proof of ergodicity by an induction
on to the number of cylinders.
Fix a finite \scs $\Sigma$ and two cylinders: $C_{j^-}$ and $C_{j^+}$.
Denote by
$F(j^-, \Sigma, j^+)$ the subset of phase points $x \in \bom$ for which
$S^{\br_-}x$ avoids the cylinder $C_{j^-}$, and there exists a $t >0$
such that $\Sigma(S^{(0, t)}x)=\Sigma$, and, moreover, $S^{t+\br_+}x$
avoids the cylinder $C_{j^+}$.
\proclaim {Conjecture 9.8} For an arbitrary \scs $\Sigma$ and any pair
of cylinders $C_{j^-}$, $C_{j^+}$, the set $F(j^-, \Sigma, j^+)$ is a closed
zero-set of codimension two.
\endproclaim
This conjecture generalizes Conjecture 8.4 and its eventual proof has an
analogous difficulty as of that one.
\heading
{\bf III.Related Directions}
\endheading
\heading {10. Replacing Topological Dimension by the Hausdorff-One}
\endheading
Let $T: \bom \to \bom$ be a transitive Anosov
$C^2$-diffeomorphism of a compact
Riemannian manifold $\bom$. Our ball-avoiding theorems discussed
so far expressed the
fact that for an orbit to be non-dense is an atypical behaviour, at least as
far as the notion of dimension we are considering is the topological one.
Surprisingly enough, if we take Hausdorff dimension, then we can not recover
this atypicality as this will be shown by the following selection of theorems.
\proclaim {Theorem 10.1\ \ [U(1991)]} Let $G$ be a non-empty open subset in
$\bom$. Then
$$
\text {\rm HD}(G\cap ND) = \dim \bom
$$
where
$$
ND= ND(T):=\ \{x \in \bom:\ T^{\bz}x\ \text {is not everywhere dense
in} \ \bom\}
$$
and $\text {\rm HD}$ denotes Hausdorff-dimension.
\endproclaim
Urba\'nski has also established an analogous statement for Anosov-flows.
\proclaim {Theorem 10.2\ \ [U(1991)]} Let $G$ be a non-empty open subset in
$\bom$. Then
$$
\text {\rm HD}(G\cap ND) = \dim \bom
$$
where $ND=ND(S)$ is the set from (1.1).
\endproclaim
Dolgopyat has found an interesting strengthening of the question answered by
the previous theorems. Note that $ND$ is the set of orbits whose limit points do
not fill up the whole space. For $Z \subset \bom$, a fixed subset we can
consider the set
$$
L_Z:= \{ x\in \bom:\ \lim T^\Bbb Z x \cap Z = \emptyset\}
$$
where $\lim T^\Bbb Z x$
denotes the set of limit points of the orbit $\{T^nx:\ t
\in \Bbb Z\}$. Dolgopyat's theorem sounds as follows:
\proclaim {Theorem 10.3\ \ [D(1997)]}
Assume $T$ is a topologically transitive Anosov
$C^2$-diffeomorphism of $\bt^2$, the two-torus, and denote by $\text {\rm HD}(\mu)$ the
Hausdorff-dimension of its Sinai-Ruelle-Bowen measure $\mu$. If $Z \subset
\bt^2$ has Hausdorff-dimension less than $\text {\rm HD}(\mu)$, then
$$
\text {\rm HD}(L_Z)=2.
$$
Conversely, for any $p > \text {\rm HD}(\mu)$, one can find a set $Z$ of
Hausdorff-dimension less than $p$ for which the above statement fails.
\endproclaim
The proofs of theorems 10.1-3 all exploit the existence of a finite
Markov partition. Furthermore, the verifications of Theorems 10.1-2 use
a generalization of a result of McMullen, [McM(1987)] providing a lower
bound for the Hausdorff dimension through local densities. On the other
hand, for establishing theorem 10.3, Dolgopyat uses formulas of
Manning-McCluskey, [MM(1983)] and Young, [Y(1982)] which are valid
in the two-dimensional
setting and this fact explains the dimensional restriction in Theorem
10.3.
For systems with singularities the Markov partition, even if it exists,
can not be finite and the previous methods do not work. Nevertheless it
is reasonable to expect
\proclaim {Conjecture 10.4} Theorems 10.1-3 are valid for Anosov systems
with singularities (for the axioms of these systems see [Y(1998)] or
[Ch(1998)]).
\endproclaim
Finally we note that results analogous to the aforementioned theorems
have been formulated for certain one-parameter subgroups of some
Lie-groups but even their listing would go beyond the scope of the
present survey. For results and conjectures we refer to [M(1990)] and
[K(1998)] and we just note that in these cases again the method of
Markov partitions is not at hand but one can exploit the rich algebraic
structure instead.
\heading {11. Ball-Avoiding in Physics: Open Systems and Repellers}
\endheading
\bigskip
For a better understanding of the pre-turbulent behaviour of the Lorenz-model,
in 1979, Pianigiani and Yorke, [P-Y(1979)] initiated the study
of open dynamical systems. One main model they suggested was a dispersive
billiard with a hole. Since it is close to our basic object, let us look at
the questions they raise for this model.
Assume we are given a dispersive billiard in $\q$ and a small hole is cut
in the table. Whenever the billiard particle enters the hole, it gets absorbed
with its orbit deleted from the phase space. We select the hole to be an open
subset $B$ of the phase space and assume that the initial phase point is given
by a measure $m_0$. Let then
$$
p^+(t):= m_0\{S^{[0,t]}x \cap B = \emptyset\}
$$
be the probability that
the particle stays on the table for at least time $t$,
and
$$
p^+_A(t):= m_0\{S^{[0,t]}x \cap B = \emptyset\ \text {and}\ S^tx\in A\}
$$
the probability that it is in the set $A$ in time $t$.
\noindent {\bf Question 1.} What is the rate $p^+(t)$ converges to $0$ with,
when $t \to \infty$?
\noindent {\bf Question 2.} Does the weak limit of the conditional measure
$$
\lim_{t \to \infty} \frac{p^+_A(t)}{p^+(t)}= \mu^+\{A\}
$$
exist and if it does what is its value?
\noindent {\bf Question 3.} How does $\mu^+$ depend on the initial distribution
$m_0$?
The questions can be raised in a time-symmetric way, too. Indeed, denote
$$
p(t):= m_0\{S^{[-t,t]}x \cap B = \emptyset\}
$$
and
$$
p_A(t):= m_0\{S^{[-t,t]}x \cap B = \emptyset\ \text {and}\ S^tx\in A\}
$$
$$
\mu\{A\}= \lim_{t \to \infty} \frac{p_A(t)}{p(t)}.
$$
Then we can pose the same questions for these objects as before.
Pianigiani and Yorke answered questions 1-3 for expanding maps acting in a
domain of $\br^d$. In a recent work of Chernov, Markarian and Troubetzkoy,
[Ch-M-T(1998)] the problems are settled for Anosov diffeomorphisms on surfaces
with small holes. Their and previous rigorous results of other authors
have been based on
analytic calculations obtained originally
by physicists. Out of these --- without aiming at completeness ---
we only mention the works of Kantz and Grassberger, [K-G(1985)]
(related, in particular, to Theorem 11.5
below), of Hsu, Ott and Grebogi, [H-O-G(1988)]
(related, in particular, Theorem 11.1 below) and
of Legrand and Sornette, [L-S(1990)]
(as to an analytic calculation for stadia); for a review we refer to
the survey of T\'el, [T(1996)].
Since we only plan to give the flavour of the results of [Ch-M-T(1998)],
we will omit the very technical formulation of their conditions.
Let $T:\ \bom \to \bom$ be a topologically transitive Anosov
$C^{1+\alpha}$-diffeomorphism of a compact Riemannian surface and $B \subset
\bom $ be a nice open subset. Denote $\widetilde \bom:= \bom
\setminus B$ and let for
every $n \ge 0$
$$
\bom_n := \cap_{i=0}^n T^i\widetilde \bom\ \ \ \text {and}\ \ \
\bom_{-n} := \cap_{i=0}^n T^{-i}\widetilde \bom
$$
and, moreover,
$$
\bom_+ := \cap_{n\ge 1} \bom_n, \qquad
\bom_- := \cap_{n\ge 1} \bom_{-n}, \qquad \Omega := \bom_-\cap \bom_+.
$$
The set $\Omega$ is called {\it the repeller} (in physics literature, recently
they are often called {\it chaotic saddles}).
Some more notations: for every finite Borel-measure $m$ we denote
$\vert m \vert = m\{ \bom \}$,
$$
(T_*m)\{A\} = m \{T^{-1}(A \cap \bom_1)\} \qquad\qquad\qquad (A \subset
\widetilde
\bom)
$$
$$
T_+m:=\frac{1}{\vert T_*m\vert}T_*m \qquad\qquad\qquad \text {if}\ \
\vert T_*m\vert \neq 0.
$$
We say that the probability measure $m$
on $\widetilde \bom$ is {\it conditionally
invariant } under $T$ if $T_+m=m$, or equivalently if there is a $\lambda_+ >0$
such that $T_*m=\lambda_+ m$. Any conditionally invariant measure $m$ is, of
course, supported on $\bom_+$, and we also have $\lambda_+
= \vert T_*m\vert = m\{\bom_{-1} \cap \bom_+\} = m\{\bom_{-1}\}$.
Denote by $\Cal M_n$, $\Cal M_+$ and $\Cal M$
the classes of (SRB-like)
probability measures supported on $\bom_n$, $\bom_+$ and
$\Omega$, respectively.
\proclaim {Theorem 11.1\ \ [Ch-M-T(1998)]} There is a unique (SRB-like) conditionally
invariant measure $\mu_+ \in \Cal M_+$, i. e. the operator $T_+ :
\Cal M_+ \to \Cal M_+$ has a unique fixed point $\mu_+$.
\endproclaim
\proclaim {Theorem 11.2\ \ [Ch-M-T(1998)]} For any measure $m_0\in \Cal M_0$, the sequence
of measures $T_+^nm_0$ converges weakly, as $n \to \infty$, to the
conditionally invariant measure $\mu_+$. Moreover, the sequence of measures
$\lambda_+^{-n} (T_*^nm_0)$ converges weakly to $\rho(m_0)\mu_+$, where the
functions $\rho(m_0)$ and $\rho^{-1}(m_0)$ are
uniformly bounded on $\Cal M_0$.
\endproclaim
\proclaim {Theorem 11.3\ \ [Ch-M-T(1998)]}
The sequence $T^{-n}\mu_+$ converges weakly, as
$n \to \infty$, to a $T$-invariant probability measure $\hat \mu_+ \in \Cal M$.
The measure $\hat \mu_+$ is ergodic and K-mixing.
\endproclaim
The aforementioned results have their natural duals by changing the signs,
and then one obtains $\mu_-, \hat \mu_-, \lambda_-$.
\proclaim {Theorem 11.4\ \ [Ch-M-T(1998)]}
If for every periodic point $x \in \Omega$, $T^kx=x$
we have $\vert \det DT^kx\vert = 1$, then $\hat \mu_+ = \hat \mu_- =
\hat \mu $ and $ \lambda_+ = \lambda_- =\lambda$.
In particular, this happens if
the given Anosov-diffeomorphism preserves a smooth invariant measure.
\endproclaim
(In [Ch-M-T(1998)] it is also conjectured that $\hat \mu_+$ is a
Bernoulli-measure, and has a fast decay of correlations.) The following
theorem not only
answers Question 1, most interesting from the point of view of
physical applications, but also proves the {\it escape rate formula} of
K-G(1988). We call $\gamma_+:= - \log \lambda_+$ the {\it
escape rate} of the system. Denote by $\lambda_+$
the positive Lyapunov exponent
of the ergodic measure $\hat \mu_+$, and by $h(\hat \mu_+)$ its
Kolmogorov-Sinai
entropy.
\proclaim {Theorem 11.5\ \ [Ch-M-T(1998)]}
$$
\gamma_+ = \lambda_+ - h(\hat \mu_+) \tag 11.6
$$
\endproclaim
An interesting feature of the escape rate formula (11.6) is that its right
hand side is defined exclusively in terms of the measure $\hat
\mu_+$ given on the
repeller $\bom_+$, whereas $\gamma_+$ is the rate with which an initial measure
given on whole $\bom$ gets pulled down to the repeller.
It is an interesting task to
generalize Theorems 11.1-5 for Anosov-systems with singularities and
subsequently for billiards.
\bigskip
\noindent {\bf Remark 11.7.} For dynamical systems of large linear size
$L$, which actually are appropriate models of transport phenomena,
Gaspard and Nicolis,
[G-N(1990)] derived a beautiful equation replacing the escape rate
formula. For definiteness, let us think of a Lorentz-process (i. e. a
dispersive, finite-horizon billiard with a periodic configuration of
scatterers) in an elongated
periodic container of integer length $L$; the boundary condition in the
direction of the $y$-axis is periodic, whereas those in the direction of the
$x$-axis at $x=0$ and $x=L$ are open, i. e. $B = (\{x=0\} \cup \{x=L\})
\times S^1$. This model determines a repeller $\bom_+(L)$
with SRB-like invariant measure $\hat \mu_+(L)$, for which we denote the
positive Lyapunov-exponent by $\lambda_L$ and the K-S entropy by
$h(\hat \mu_+(L))$.
Then, by using the diffusion approximation for the
Lorentz-process, Gaspard and Nicolis proved analytically that
$$
\Cal D = \lim_{L\to \infty} (\frac{L}{\pi^2})\left( \lambda_L -
h(\hat \mu_+(L))
\right)
$$
where $\Cal D$ is the diffusion coefficient of the Lorentz process
in the infinite slab (i. e. in the same model with $L=\infty$).
Further related formulas and models are beyond the scope of the present
survey. As references on this most alive direction of research we mention
the papers of Gaspard and Dorfman, [G-D(1995)], of T\'el, Vollmer and Breymann,
[T-V-B(1996)] and of Ruelle, [R(1999)]; for earlier related models of
transport see the
works of Lebowitz and Spohn, [L-S(1978)] and of Kr\'amli, Sim\'anyi and
Sz\'asz, [K-S-Sz(1987)].
\bigskip
\noindent {\bf Acknowledgment.} It is my pleasure to thank P\'eter B\'alint,
Andr\'as Kr\'amli and N\'andor
Sim\'anyi for their most careful reading of
the manuscript and for their critical remarks. Moreover,
I am the most grateful to Howie
Weiss and Tam\'as T\'el for several basic references related to Sections 10
and 11, respectively.
The research was also supported
by the Hungarian-Portuguese joint grant No. P/97-10 through the cooperation
of the
Hungarian National Committee for Technological Development and the Instituto
de Cooperacao Cientifica e Tecnol\'ogica Internacional.
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\enddocument