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Hard sphere gas, hard ball systems, Boltzmann hypothesis, ergodicity,
local ergodicity, non-uniform hyperbolicity, singularities, stable and unstable foliations
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\input amstex.tex
\documentstyle{amsppt}
\magnification=\magstep1
\vsize 8.5truein
\hsize 6truein
\define\flow{\left(\bold{M},\{S^t\}_{t\in\Bbb R},\mu\right)}
\define\symb{\Sigma=\left(\sigma_1,\sigma_2,\dots,\sigma_n\right)}
\define\traj{S^{[a,b]}x_0}
\define\sdb{semi-dispersing billiard}
\define\sdbs{semi-dispersing billiards}
\define\bil{billiard}
\def\boq{{\bold {Q}}}
\def\bn{{\Bbb N}}
\def\ks{\Cal S}
\def\kr{\Cal R}
\define\sr{\ks\kr^+}
\def\g{\gamma}
\define\spix{S^{(0,\infty)}x}
\define\xu{U(x_0)}
\define\yu{U(y_0)}
\define\sinf{S^{(-\infty,\infty)}x_0\,}
\define\Q{\bold {Q}}
\define\suf{sufficient\ \ }
\define\erg{ergodicity\ \ }
\define\proj{\Bbb P^{\nu-1}(\Bbb R)}
\define\ter{\Bbb S^{d-1}\times\left[\Bbb P^{\nu-1}(\Bbb R)\right]^n}
\define\sphere{\Bbb S^{d-1}}
\define\pont{(v^0;h_1,h_2,\dots ,h_n)}
\define\projn{\left[\Bbb P^{\nu-1}(\Bbb R)\right]^n}
\define\pontg{(v^0;g_1,g_2,\dots ,g_n)}
\define\szorzat{\prod\Sb i=1\endSb\Sp n\endSp \Cal P_i}
\define\ball{\overline{B}_{\varepsilon_0}(x_0)}
\define\qv{(Q,V^+)}
\noindent
May 14, 2006
\bigskip \bigskip
\heading
Conditional Proof of the Boltzmann-Sinai Ergodic Hypothesis \\
(Assuming the Hyperbolicity of Typical Singular Orbits)
\endheading
\bigskip \bigskip
\centerline{{\bf N\'andor Sim\'anyi}
\footnote{Research supported by the National Science Foundation, grant
DMS-0457168.}}
\bigskip \bigskip
\centerline{University of Alabama at Birmingham}
\centerline{Department of Mathematics}
\centerline{Campbell Hall, Birmingham, AL 35294 U.S.A.}
\centerline{E-mail: simanyi\@math.uab.edu}
\bigskip \bigskip
\noindent
{\it Dedicated to Yakov G. Sinai and Domokos Sz\'asz}
\bigskip \bigskip
\hbox{\centerline{\vbox{\hsize 8cm {\bf Abstract.} We consider the system of
$N$ ($\ge2$) elastically colliding hard balls of masses $m_1,\dots,m_N$
and radius $r$ on the flat unit torus $\Bbb T^\nu$, $\nu\ge2$. We prove
the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full
hyperbolicity and ergodicity of such systems for every selection
$(m_1,\dots,m_N;r)$ of the external geometric parameters, under the
assumption that almost every singular trajectory is geometrically
hyperbolic (sufficient), i. e. the so called Chernov-Sinai Ansatz holds
true for the model. The present proof does not use at all the formerly
developed, rather involved algebraic techniques, instead it employs
exclusively dynamical methods and tools from geometric analysis.}}}
\bigskip \bigskip
\noindent
Primary subject classification: 37D50
\medskip
\noindent
Secondary subject classification: 34D05
\bigskip \bigskip
\heading
\S1. Introduction
\endheading
\bigskip
In this paper we prove the Boltzmann--Sinai Ergodic Hypothesis under the
condition of the Chernov-Sinai Ansatz (see \S2).
In a loose form, as attributed to L. Boltzmann back in the 1880's,
this hypothesis asserts that gases of hard balls are ergodic. In a precise
form, which is due to Ya. G. Sinai in 1963 [Sin(1963)], it states that the gas
of $N\ge2$ identical hard balls (of "not too big" radius) on a torus
$\Bbb T^\nu$, $\nu\ge2$, (a $\nu$-dimensional box with periodic
boundary conditions) is ergodic, provided that certain necessary
reductions have been made. The latter means that one fixes the
total energy, sets the total momentum to zero, and restricts the
center of mass to a certain discrete lattice within the torus. The
assumption of a not too big radius is necessary to have the interior of the
configuration space connected.
Sinai himself pioneered rigorous mathematical studies of hard ball
gases by proving the hyperbolicity and ergodicity for the case
$N=2$ and $\nu=2$ in his seminal paper [Sin(1970)],
where he laid down the foundations
of the modern theory of chaotic billiards. Then Chernov and Sinai
extended this result to ($N=2$, $\nu\ge 2$), as well as proved
a general theorem on ``local'' ergodicity applicable to systems of
$N>2$ balls [S-Ch(1987)]; the latter became instrumental in the
subsequent studies. The case $N>2$ is substantially more difficult
than that of $N=2$ because, while the system of two balls reduces
to a billiard with strictly convex (spherical) boundary, which
guarantees strong hyperbolicity, the gases of $N>2$ balls reduce
to billiards with convex, but not strictly convex, boundary (the
latter is a finite union of cylinders) -- and those are characterized
by very weak hyperbolicity.
Further development has been mostly due to A. Kr\'amli,
D. Sz\'asz, and the present author. We proved hyperbolicity and
ergodicity for $N=3$ balls in any dimension [K-S-Sz(1991)] by
exploiting the ``local'' ergodic theorem of Chernov and Sinai
[S-Ch(1987)], and carefully analyzing all
possible degeneracies in the dynamics to obtain ``global''
ergodicity. We extended our results to $N=4$ balls in
dimension $\nu\ge3$ next year [K-S-Sz(1992)], and then I proved the
ergodicity whenever $N\le\nu$ [Sim(1992)-I-II] (this covers systems with an
arbitrary number of balls, but only in spaces of high enough
dimension, which is a restrictive condition). At this point,
the existing methods could no longer handle any new cases, because
the analysis of the degeneracies became overly complicated. It was
clear that further progress should involve novel ideas.
A breakthrough was made by Sz\'asz and myself, when we used
the methods of algebraic geometry [S-Sz(1999)]. We assumed that the balls had
arbitrary masses $m_1,\dots,m_N$ (but the same radius $r$). Now
by taking the limit $m_N\to 0$, we were able to reduce the
dynamics of $N$ balls to the motion of $N-1$ balls, thus utilizing
a natural induction on $N$. Then algebro-geometric methods allowed us to
effectively analyze all possible degeneracies, but only for
typical (generic) $(N+1)$-tuples of ``external'' parameters
$(m_1,\dots,m_N,r)$; the latter needed to avoid some exceptional
submanifolds of codimension one, which remained unknown. This
approach led to a proof of full hyperbolicity (but not yet
ergodicity) for all $N\ge2$ and $\nu\ge2$, and for generic
$(m_1,\dots,m_N,r)$, see [S-Sz(1999)]. Later the present author simplified
the arguments and made them more ``dynamical'', which allowed me
to obtain full hyperbolicity for hard balls with any set of
external geometric parameters
$(m_1,\dots,m_N,r)$ [Sim(2002)]. Thus, the hyperbolicity has
been fully established for all systems of hard balls on tori.
To upgrade the full hyperbolicity to ergodicity, one needs to refine the
analysis of the aforementioned degeneracies. For hyperbolicity, it was enough
that the degeneracies made a subset of codimension $\ge1$ in the phase space.
For ergodicity, one has to show that its codimension is $\ge2$, or to find
some other ways to prove that the (possibly) arising codimension-one manifolds
of non-sufficiency are incapable of separating distinct ergodic components.
The latter approach will be pursued in this paper. In the paper [Sim(2003)] I
took the first step in the direction of proving that the codimension of
exceptional manifolds is at least two: I proved that the systems of $N\ge2$
balls on a 2D torus (i.e., $\nu=2$) are ergodic for typical (generic)
$(N+1)$-tuples of external parameters $(m_1,\dots,m_N,r)$. The proof again
involves some algebro-geometric techniques, thus the result is restricted to
generic parameters $(m_1,\dots,m_N;\,r)$. But there was a good reason to
believe that systems in $\nu\ge3$ dimensions would be somewhat easier to
handle, at least that was indeed the case in early studies.
Finally, in my recent paper [Sim(2004)] I was able to further improve the
algebro-geometric methods of [S-Sz(1999)], and proved that for any $N\ge2$,
$\nu\ge2$ and for almost every selection $(m_1,\dots,m_N;\,r)$ of the external
geometric parameters the corresponding system of $N$ hard balls on
$\Bbb T^\nu$ is (fully hyperbolic and) ergodic.
\medskip
In this paper I will prove the following result.
\medskip
\subheading{Theorem} For any integer values $N\ge2$, $\nu\ge2$,
and for every $(N+1)$-tuple $(m_1,\dots,m_N,r)$ of the external geometric
parameters the standard hard ball system
$\left(\bold M_{\vec m,r},\,\left\{S_{\vec m,r}^t\right\},\,
\mu_{\vec m,r}\right)$ is (fully hyperbolic and) ergodic, provided that the
so Chernov-Sinai Ansatz (see the closing part of \S2 below) is true
for $\flow$ and for all of its subsystems.
\medskip
\subheading{Remark 1.1} The novelty of the theorem (as compared to the result
in [Sim(2004)]) is that it applies to each $(N+1)$-tuple of external
parameters (provided that the interior of the phase space is connected),
without an exceptional zero-measure set.
\medskip
\subheading{Remark 1.2} The present result speaks about exactly the same
models as the result of [Sim(2002)], but the assertion of this new theorem is
obviously stronger than that of the theorem in [Sim(2002)]: It has been known
for a long time that, for the family of semi-dispersive billiards, ergodicity
cannot be obtained without also proving full hyperbolicity.
\medskip
\subheading{Remark 1.3} As it follows from the results of [C-H(1996)] and
[O-W(1998)], all standard hard ball systems $\flow$ (the models covered by the
theorem) are not only ergodic, but they enjoy the Bernoulli mixing property,
as long as they are known to be ergodic.
\medskip
\subheading{Remark 1.4} The reason for assuming the Ansatz not only for the
considered model $\flow$ but also for all of its subsystems is the inductive
nature of the proof, see \S5.
\bigskip
\subheading{The Organization of the Paper}
In the subsequent section we overview
the necessary technical prerequisites of the proof, along with the needed
references to the literature. The fundamental objects of this paper are the so
called "exceptional $J$-manifolds": they are codimension-one
submanifolds of the phase space that are separating distinct, open
ergodic components of the billiard flow. In \S3 we obtain the
necessary linear estimations for the contraction coefficients of some specific
tangent vectors in small, tubular neighborhoods of exceptional manifolds.
By using the results of \S3, in \S4 we prove that at least one phase
point of an exceptional $J$-manifold is actually sufficient (Main
Lemma 4.5).
Finally, in the closing section we complete the inductive proof of ergodicity
(with respect to the number of balls $N$) by utilizing Main Lemma 4.5 and
earlier results from the literature. Actually, a consequence of Main Lemma
4.5 will be that exceptional $J$-manifolds do not exist, and this will imply
the the fact that no distinct, open ergodic components can coexist.
Finally, a short appendix of this paper serves the purpose of making the
reading of the proof of \S4 easier, by providing a chart of the hierarchy of
the selection of several constants playing a role in the proof of
Main Lemma 4.5.
\bigskip \bigskip
\heading
\S2. Prerequisites
\endheading
\bigskip
Consider the $\nu$-dimensional ($\nu\ge2$), standard, flat torus
$\Bbb T^\nu=\Bbb R^\nu/\Bbb Z^\nu$ as the vessel containing
$N$ ($\ge2$) hard balls (spheres) $B_1,\dots,B_N$ with positive masses
$m_1,\dots,m_N$ and (just for simplicity) common radius $r>0$. We always
assume that the radius $r>0$ is not too big, so
that even the interior of the arising configuration space $\bold Q$ (or,
equivalently, the phase space) is connected. Denote the center of the ball
$B_i$ by $q_i\in\Bbb T^\nu$, and let $v_i=\dot q_i$ be the velocity of the
$i$-th particle. We investigate the uniform motion of the balls
$B_1,\dots,B_N$ inside the container $\Bbb T^\nu$ with half a unit of total
kinetic energy: $E=\dfrac{1}{2}\sum_{i=1}^N m_i||v_i||^2=\dfrac{1}{2}$.
We assume that the collisions between balls are perfectly elastic. Since
--- beside the kinetic energy $E$ --- the total momentum
$I=\sum_{i=1}^N m_iv_i\in\Bbb R^\nu$ is also a trivial first integral of the
motion, we make the standard reduction $I=0$. Due to the apparent translation
invariance of the arising dynamical system, we factorize the configuration
space with respect to uniform spatial translations as follows:
$(q_1,\dots,q_N)\sim(q_1+a,\dots,q_N+a)$ for all translation vectors
$a\in\Bbb T^\nu$. The configuration space $\bold Q$ of the arising flow
is then the factor torus
$\left(\left(\Bbb T^\nu\right)^N/\sim\right)\cong\Bbb T^{\nu(N-1)}$
minus the cylinders
$$
C_{i,j}=\left\{(q_1,\dots,q_N)\in\Bbb T^{\nu(N-1)}\colon\;
\text{dist}(q_i,q_j)<2r \right\}
$$
($1\le i0$ is the cosine of the
angle $\phi$ ($0\le\phi<\pi/2$) subtended by $v_t^+$ and $\nu(q_t)$. Regarding
formulas (3.3), please see the last displayed formula in \S1 of
[S-Ch(1987)] or (i)--(ii) in Proposition 2.3 of [K-S-Sz(1990)-I]. The
instanteneous change in the infinitesimal Lyapunov function
$Q(\delta q_t,\delta v_t)$ caused by the reflection at time $t>0$ is easily
derived from (3.3):
$$
\aligned
Q(\delta q_t^+,\delta v_t^+)&=Q(\delta q_t^-,\delta v_t^-)+2\cos\phi\langle
V\delta q_t^-,\, KV\delta q_t^-\rangle \\
&\ge Q(\delta q_t^-,\delta v_t^-).
\endaligned
\tag 3.4
$$
In the last inequality we used the fact that the operator $K$ is positive
semi-definite, i. e. the billiard is semi-dispersive.
We are primarily interested in getting useful lower estimates for the expansion
rate $||\delta q_t||/||\delta q_0||$. The needed result is
\medskip
\subheading{Proposition 3.5} Use all the notations above, and assume that
$$
\langle\delta q_0,\, \delta v_0\rangle/||\delta q_0||^2\ge c_0>0.
$$
We claim that $||\delta q_t||/||\delta q_0||\ge 1+c_0t$ for all $t\ge0$.
\medskip
\subheading{Proof} Clearly, the function $||\delta q_t||$ of $t$ is continuous
for all $t\ge0$ and continuously differentiable between collisions. According
to (3.1), $\frac{d}{dt}\delta q_t=\delta v_t$, so
$$
\frac{d}{dt}||\delta q_t||^2=2\langle\delta q_t, \delta v_t\rangle.
\tag 3.6
$$
Observe that not only the positive valued function
$Q(\delta q_t,\delta v_t)=\langle\delta q_t,\delta v_t\rangle$ is
nondecreasing in $t$ by (3.2) and (3.4), but the quantity
$\langle\delta q_t,\delta v_t\rangle/||\delta q_t||$ is nondecreasing in $t$,
as well. The reason is that
$\langle\delta q_t,\delta v_t\rangle/||\delta q_t||=||\delta v_t||\cos\alpha_t$
($\alpha_t$ being the acute angle subtended by $\delta q_t$ and $\delta v_t$),
and between collisions the quantity $||\delta v_t||$ is unchanged, while the
acute angle $\alpha_t$ decreases, according to the time-evolution equations
(3.1). Finally, we should keep in mind that at a collision the norm
$||\delta q_t||$ does not change, while $\langle\delta q_t,\delta v_t\rangle$
cannot decrease, see (3.4). Thus we obtain the inequalities
$$
\langle\delta q_t,\delta v_t\rangle/||\delta q_t||\ge
\langle\delta q_0,\delta v_0\rangle/||\delta q_0||\ge c_0||\delta q_0||,
$$
so
$$
\frac{d}{dt}||\delta q_t||^2=2||\delta q_t||\frac{d}{dt}||\delta q_t||
=2\langle\delta q_t,\delta v_t\rangle
\ge 2c_0||\delta q_0||\cdot||\delta q_t||
$$
by (3.6). This means that
$\frac{d}{dt}||\delta q_t||\ge c_0||\delta q_0||$, so
$||\delta q_t||\ge||\delta q_0||(1+c_0t)$, proving the proposition. \qed
\medskip
Next we need an effective lower estimation $c_0$ for the curvature
$\langle\delta q_0,\, \delta v_0\rangle/||\delta q_0||^2$ of the trajectory
bundle:
\medskip
\subheading{Lemma 3.7} Assume that the perturbation
$(\delta q_0^-,\,\delta v_0^-)\in\Cal T_{x_0}\bold M$ (as in Proposition 3.5)
is being performed at time zero right before a collision, say,
$\sigma_0=(1,\,2)$ taking place at that time. Select the tangent vector
$(\delta q_0^-,\,\delta v_0^-)$ in such a specific way that $\delta v_0^-=0$,
$\delta q_0^-=(m_2w,-m_1w,0,0,\dots,0)$ with a nonzero vector $w\in\Bbb R^\nu$,
$\langle w,v_1^--v_2^-\rangle=0$. This scalar product equation is exactly the
condition that guarantees that $\delta q_0^-$ be orthogonal to the velocity
component $v^-=(v_1^-,v_2^-,\dots,v_N^-)$ of $x_0=(q,v^-)$. The last, though
crucial requirement is that $w$ should be selected from the two-dimensional
plane spanned by $v_1^--v_2^-$ and $q_1-q_2$ (with $||q_1-q_2||=2r$) in
$\Bbb R^\nu$. The purpose of this condition is to avoid the unwanted
phenomenon of ``astigmatism'' in our billiard system, discovered first by
Bunimovich and Rehacek in [B-R(1997)] and [B-R(1998)]. Later on the phenomenon
of astigmatism gathered further prominence in the paper [B-Ch-Sz-T(2002)] as
the main driving mechanism behind the wild non-differentiability of the
singularity manifolds (at their boundaries) in hard ball systems in dimensions
bigger than $2$. We claim that
$$
\frac{\langle\delta q_0^+,\delta v_0^+\rangle}{||\delta q_0||^2}
=\frac{||v_1-v_2||}{r\cos\phi_0}\ge\frac{||v_1-v_2||}{r}
\tag 3.8
$$
for the post-collision tangent vector $(\delta q_0^+,\delta v_0^+)$, where
$\phi_0$ is the acute angle subtended by $v_1^+-v_2^+$ and the outer normal
vector of the sphere $\left\{y\in\Bbb R^\nu\big|\; ||y||=2r\right\}$ at the
point $y=q_1-q_2$. Note that in (3.8) there is no need to use $+$ or $-$ in
$||\delta q_0||^2$ or $||v_1-v_2||$, for $||\delta q_0^-||=||\delta q_0^+||$,
$||v_1^--v_2^-||=||v_1^+-v_2^+||$.
\medskip
\subheading{Proof} The proof of the equation in (3.8) is a simple, elementary
geometric argument in the plane spanned by $v_1^--v_2^-$ and $q_1-q_2$, so we
omit it. We only note that the outgoing relative velocity $v_1^+-v_2^+$ is
obtained from the pre-collision relative velocity $v_1^--v_2^-$ by reflecting
the latter one across the tangent hyperplane of the sphere
$\left\{y\in\Bbb R^\nu\big|\; ||y||=2r\right\}$ at the point $y=q_1-q_2$. \qed
\medskip
The previous lemma shows that, in order to get useful lower estimations for the
``curvature'' $\langle\delta q,\delta v\rangle/||\delta q||^2$ of the
trajectory bundle, it is necessary
(and sufficient) to find collisions $\sigma=(i,j)$ on the orbit of a given
point $x_0\in\bold M$ with a ``relatively big'' value of $||v_i-v_j||$.
Finding such collisions will be based upon the following result:
\medskip
\subheading{Proposition 3.9} Consider orbit segments $S^{[0,T]}x_0$ of
$N$-ball systems with masses $m_1,m_2,\dots,m_N$ in $\Bbb T^\nu$ (or in
$\Bbb R^\nu$) with collision sequences
$\Sigma=(\sigma_1,\sigma_2,\dots,\sigma_n)$ corresponding to connected
collision graphs. (Now the kinetic energy is not necessarily normalized, and
the total momentum $\sum_{i=1}^N m_iv_i$ may be different from zero.) We claim
that there exists a positive-valued function $f(a;m_1,m_2,\dots,m_N)$
($a>0$, $f$ is independent of the orbit segments $S^{[0,T]}x_0$) with the
following two properties:
\medskip
(1) If $||v_i(t_l)-v_j(t_l)||\le a$ for all collisions $\sigma_l=(i,j)$
($1\le l\le n$, $t_l$ is the time of $\sigma_l$) of some trajectory segment
$S^{[0,T]}x_0$ with a symbolic collision sequence
$\Sigma=(\sigma_1,\sigma_2,\dots,\sigma_n)$ corresponding to a connected
collision graph, then the norm $||v_{i'}(t)-v_{j'}(t)||$ of any relative
velocity at any time $t\in\Bbb R$ is at most $f(a;m_1,\dots,m_N)$;
(2) $\lim_{a\to 0} f(a;m_1,\dots,m_N)=0$ for any $(m_1,\dots,m_N)$.
\medskip
\subheading{Proof} We begin with
\medskip
\subheading{Lemma 3.10} Consider an $N$-ball system with masses $m_1,\dots,m_N$
(an $(m_1,\dots\allowmathbreak,m_N)$-system, for short) in $\Bbb T^\nu$
(or in $\Bbb R^\nu$).
Assume that the inequalities $||v_i(0)-v_j(0)||\le a$ hold true
($1\le i0$ (depending only on $N$, $m_1,\dots,m_N$)
with the following property:
In any orbit segment $S^{[0,T]}x_0$ of the $(m_1,\dots,m_N)$-system with the
standard normalizations and with a connected collision graph, one can always
find a collision $\sigma=(i,j)$, taking place at time $t$, so that
$||v_i(t)-v_j(t)||\ge G(m_1,\dots,m_N)$.
\medskip
\subheading{Proof} Indeed, we choose $G=G(m_1,\dots,m_N)>0$ so small that
$f(G;m_1,\dots,m_N)\allowmathbreak0$
one can always find a time $-t<0$ and a tangent vector
$(\delta q_0,\delta v_0)\in\Cal T_{x_{-t}}\bold M$ ($x_{-t}=S^{-t}x_0$) with
$\langle \delta q_0,\delta v_0\rangle>0$ and
$||\delta q_t||/||\delta q_0||>L$, where
$(\delta q_t,\delta v_t)=DS^t(\delta q_0,\delta v_0)$.
\medskip
\subheading{Proof} Indeed, select a number $t>0$ so big that
$1+\frac{t}{r}G(m_1,\dots,m_N)>L$ and $-t$ is the time of a collision (on the
orbit of $x_0$) with the relative velocity $v^-_i(-t)-v^-_j(-t)$, for which
$||v^-_i(-t)-v^-_j(-t)||\ge G(m_1,\dots,m_N)$. By Lemma 3.7 we can choose a
tangent vector $(\delta q^-_0,\,0)$ right before the collision at time
$-t$ in such a way that the lower estimation
$$
\frac{\langle\delta q^+_0,\,\delta v^+_0\rangle}{||\delta q^+_0||^2}\ge
\frac{1}{r}G(m_1,\dots,m_N)
$$
holds true for the ``curvature''
$\langle\delta q^+_0,\,\delta v^+_0\rangle/||\delta q^+_0||^2$ associated
with the post-collision tangent vector $(\delta q^+_0,\,\delta v^+_0)$.
According to Proposition 3.5 we have then the lower estimation
$$
\frac{||\delta q_t||}{||\delta q_0||}\ge 1+\frac{t}{r}G(m_1,\dots,m_N)>L
$$
for the $\delta q$-expansion rate between $(\delta q^-_0,\,0)$ and
$(\delta q_t,\,\delta v_t)=DS^t(\delta q^-_0,\,0)$. \qed
\bigskip \bigskip
\heading
\S4. The Exceptional $J$-Manifolds \\
(The asymptotic measure estimates)
\endheading
\bigskip \bigskip
First of all, we define the fundamental object for the proof of our theorem.
\medskip
\subheading{Definition 4.1} A smooth submanifold $J\subset\text{int}\bold M$
of the interior of the phase space $\bold M$ is called an {\it exceptional
$J$-manifold} (or simply an exceptional manifold) with a negative Lyapunov
function $Q$ if
\medskip
(1) $\text{dim}J=2d-2$ ($=\text{dim}\bold M-1$);
\medskip
(2) the pair of manifolds $(\overline{J},\,\partial J)$ is diffeomorphic
to the standard pair
$$
\left(B^{2d-2},\,\Bbb S^{2d-3}\right)=\left(B^{2d-2},\,
\partial B^{2d-2}\right),
$$
where $B^{2d-2}$ is the closed unit ball of $\Bbb R^{2d-2}$;
\medskip
(3) $J$ is locally flow-invariant, i. e. $\forall x\in J$
$\exists\,a(x),\,b(x)$, $a(x)<0**0$ such that
(i) $S^T\left(\tilde{U}_0\right)\cap\partial\bold M=\emptyset$, and
all orbit segments $S^{[0,T]}x$ ($x\in\tilde U_0$) are non-singular, hence
they share the same symbolic collision sequence $\Sigma$;
(ii) $\forall x\in\tilde U_0$ the orbit segment $S^{[0,T]}x$ is sufficient
if and only if $x\not\in J$;
\medskip
(5) $\forall x\in J$ we have $Q(n(x)):=\langle z(x),\,w(x)\rangle\le -c_1<0$
for a unit normal vector field $n(x)=(z(x),\,w(x))$ of $J$ with a fixed
constant $c_1>0$;
\medskip
(6) the set $W$ of phase points $x\in J$ never again returning to $J$ (After
first leaving it, of course. Keep in mind that $J$ is locally flow-invariant!)
has relative measure greater than $1-10^{-8}$ in $J$, i. e.
$\dfrac{\mu_1(W)}{\mu_1(J)}>1-10^{-8}$, where $\mu_1$ is the hypersurface
measure of the smooth manifold $J$.
\bigskip
We begin with an important proposition on the structure of forward orbits
$S^{[0,\infty)}x$ for $x\in J$.
\medskip
\subheading{Proposition 4.2} For $\mu_1$-almost every $x\in J$
the forward orbit $S^{[0,\infty)}x$ is non-singular.
\medskip
\subheading{Proof} According to Proposition 7.12 of [Sim(2003)], the set
$$
J\cap\left[\bigcup\Sb t>0\endSb S^{-t}\left(\Cal S\Cal R^-\right)\right]
$$
of forward singular points $x\in J$ is a countable union of smooth, proper
submanifolds of $J$, hence it has $\mu_1$-measure zero. \qed
\medskip
In the future we will need
\medskip
\subheading{Lemma 4.3} The concave, local orthogonal manifolds $\Sigma(y)$
passing through points $y\in J$ are uniformly transversal to $J$.
\medskip
\subheading{Note} A local orthogonal manifold $\Sigma\subset\text{int}\bold M$
is obtained from a codimension-one, smooth submanifold $\Sigma_1$ of
$\text{int}\bold Q$ by supplying $\Sigma_1$ with a selected field of unit
normal vectors as velocities. $\Sigma$ is said to be concave if the second
fundamental form of $\Sigma_1$ (with respect to the selected orientation)
is negative semi-definite at every point of $\Sigma_1$. Similarly, the
convexity of $\Sigma$ requires positive semi-definiteness here,
see also \S2 of [K-S-Sz(1990)-I].
\medskip
\subheading{Proof} We will only prove the transversality. It will
be clear from the uniformity of the estimations used in the proof that the
claimed transversalities are actually uniform across $J$.
Assume, to the contrary of the transversality, that a concave, local orthogonal
manifold $\Sigma(y)$ is tangent to $J$ at some $y\in J$. Let
$(\delta q,\,B\delta q)$ be any vector of $\Cal T_y\bold M$ tangent to
$\Sigma(y)$ at $y$. Here $B\le 0$ is the second fundamental form of the
projection $q\left(\Sigma(y)\right)=\Sigma_1(y)$ of $\Sigma(y)$ at the point
$q=q(y)$. The assumed tangency means that
$\langle\delta q,\,z\rangle+\langle B\delta q,\,w\rangle=0$, where
$n(y)=\left(z(y),\,w(y)\right)=(z,w)$ is the unit normal vector of $J$ at $y$.
We get that $\langle\delta q,\,z+Bw\rangle=0$ for any vector
$\delta q\in v(y)^\perp$. We note that the components $z$ and $w$ of $n$ are
necessarily orthogonal to the velocity $v(y)$, because the manifold $J$ is
locally flow-invariant. The last equation means that $z=-Bw$, thus
$Q(n(y))=\langle z,\,w\rangle=\langle -Bw,\,w\rangle\ge0$, contradicting to
the assumption $Q(n(y))\le -c_1$ of (5) in 4.1. This finishes the proof of
the lemma. \qed
\medskip
In order to formulate the main result of this section, we need to define two
important subsets of $J$.
\medskip
\subheading{Definition 4.4} Let
$$
A=\left\{x\in J\big|\; S^{[0,\infty)}x\text{ is nonsingular and }
\text{dim}\Cal N_0\left(S^{[0,\infty)}x\right)=1 \right\},
$$
$$
B=\left\{x\in J\big|\; S^{[0,\infty)}x\text{ is nonsingular and }
\text{dim}\Cal N_0\left(S^{[0,\infty)}x\right)>1\right\}.
$$
The two Borel subsets $A$ and $B$ of $J$ are disjoint and, according to
Proposition 4.2 above, their union $A\cup B$ has full $\mu_1$-measure in $J$.
The anticipated main result of this section is
\medskip
\subheading{Main Lemma 4.5} Use all of the above definitions and notations.
We claim that $A\ne\emptyset$.
\medskip
\subheading{Proof} The proof will be a proof by contradiction, and it will be
subdivided into several lemmas. Thus, from now on, we assume that
$A=\emptyset$.
First, select and fix a non-periodic point (a ``base point'') $x_0\in B$.
The following step is to use Corollary 3.13 in such a way that the role of
$x_0$ in 3.13 is now played by the time-inverted version $-x_0=(q_0,\,-v_0)$
of our fixed base point $x_0\in J$. Thus, for a large constant $L_0>>1$
(to be specified later) select a big enough time $c_3>>1$ (playing the role
of $t$ in 3.13) and a tangent vector
$(\delta q_0,\,-\delta v_0)\in\Cal T_{-x_{c_3}}\bold M$
($x_{c_3}=S^{c_3}x_0$) with $\langle\delta q_0,\,-\delta v_0\rangle>0$ and
$$
\frac{||\delta q_{c_3}||}{||\delta q_{0}||}>L_0,
\tag 4.6
$$
where
$$
(\delta q_{c_3},\,\delta v_{c_3})=\left(DS^{-c_3}\right)
(\delta q_0,\,\delta v_0).
\tag 4.7
$$
The normalized tangent vector
$$
(\delta\tilde q_0,\,\delta\tilde v_0):=\left(||\delta q_{c_3}||^2
+||\delta v_{c_3}||^2\right)^{-1/2}\cdot(\delta q_{c_3},\,\delta v_{c_3})
\in\Cal T_{x_0}\bold M
\tag 4.8
$$
will play a crucial role in the proof.
By slightly perturbing the tangent vector $(\delta q_0,\,-\delta v_0)$,
we can always achieve that
$$
\cases
&\text{the unit tangent vector }(\delta\tilde q_0,\,\delta\tilde v_0)
\text{ of (4.8)} \\
&\text{is transversal to } J \text{ and, corresponding to (4.6),} \\
&\frac{||\delta\tilde q_{c_3}||}{||\delta\tilde q_{0}||}0$
as the set of all phase points $\gamma_x(s)$, where $x\in J$,
$0\le s<\delta$. Here $\gamma_x(\,.\,)$ is the geodesic line passing through
$x$ (at time zero) with the initial velocity $n(x)$, $x\in J$. The radius
(thickness) $\delta>0$ here is a variable,
which will eventually tend to zero. We are
interested in getting useful asymptotic estimates for certain subsets of
$U_\delta$, as $\delta\to 0$.
Our main working domain will be the set
$$
\aligned
D_0=\Big\{&y\in U_{\delta_0}\setminus J\big|\; y\not\in\bigcup\Sb t>0\endSb
S^{-t}\left(\Cal S\Cal R^-\right),\;\exists\text{ a sequence} \\
&t_n\nearrow\infty\text{ such that }
S^{t_n}y\in U_{\delta_0}\setminus J, \quad n=1,2,\dots\Big\},
\endaligned
\tag 4.10
$$
a set of full $\mu$-measure in $U_{\delta_0}$. We will use the shorthand
notation $U_0=U_{\delta_0}$ for a fixed, small value $\delta_0$ of
$\delta$. For any $y\in\bold M$ we use the traditional notations
$$
\aligned
&\tau(y)=\min\left\{t>0\big|\; S^ty\in\partial\bold M\right\}, \\
&T(y)=S^{\tau(y)}y
\endaligned
\tag 4.11
$$
for the first hitting of the collision space $\partial\bold M$. The first
return map (Poincar\'e section, collision map)
$T:\,\partial\bold M\to\partial\bold M$ (the restriction of the above $T$
to $\partial\bold M$) is known to preserve the finite measure $\nu$ that can
be obtained from the Liouville measure $\mu$ by projecting the latter one onto
$\partial\bold M$ along the flow. Following 4. of [K-S-Sz(1990)-II], for any
point $y\in\text{int}\bold M$ (with $\tau(y)<\infty$, $\tau(-y)<\infty$, where
$-y=(q,-v)$ for $y=(q,v)$) we denote by $z_{tub}(y)$ the supremum of all radii
$\rho>0$ of tubular neighborhoods $V_\rho$ of the projected segment
$$
q\left(\left\{S^ty\big|\; -\tau(-y)\le t \le\tau(y)\right\}\right)
\subset\bold Q
$$
for which even the closure of the set
$$
\left\{(q,\,v(y))\in\bold M\big|\; q\in V_\rho\right\}
$$
does not intersect the set $\Cal S\Cal R$ of singular reflections.
We remind the reader that both Lemma 2 of [S-Ch(1987)] and Lemma 4.10 of
[K-S-Sz(1990)-I] use this tubular distance function $z_{tub}(\,.\,)$ (despite
the notation $z(\,.\,)$ in those papers), see the important note 4. on
[K-S-Sz(1990)-II].
Following the fundamental construction of local stable invariant manifolds
[S-Ch(1987)] (see also \S5 of [K-S-Sz(1990)-I]), for any $y\in D_0$ we define
the concave, local orthogonal manifolds
$$
\aligned
&\Sigma_t^t(y)=SC_{y_t}\left(\left\{(q,\,v(y_t))\in\bold M\big|\;
q-q(y_t)\perp v(y_t)\right\}\setminus(\Cal S_1\cup\Cal S_{-1})\right), \\
&\Sigma_0^t(y)=SC_y\left[S^{-t}\Sigma_t^t(y)\right],
\endaligned
\tag 4.12
$$
where $\Cal S_1:=\left\{x\in\bold M\big|\; Tx\in\Cal S\Cal R^-\right\}$
(the set of phase points on singularities of order $1$),
$\Cal S_{-1}:=\left\{x\in\bold M\big|\; -x\in\Cal S_1\right\}$
(the set of phase points on singularities of order $-1$),
$y_t=S^ty$, and $SC_y(\,.\,)$ stands for taking the smooth component of
the given set that contains the point $y$. The local, stable invariant
manifold $\gamma^{(s)}(y)$ of $y$ is known to be a superset of the
$C^2$-limiting manifold $\lim_{t\to\infty}\Sigma_0^t(y)$.
On all these local orthogonal manifolds, appearing in the proof, we will
always use the so called $\delta q$-metric to measure distances. The
length of a smooth curve with respect to this metric is the integral of
$||\delta q||$ along the curve. The proof of the Theorem on Local Ergodicity
[S-Ch(1987)] shows that the $\delta q$-metric is the relevant notion of
distance on the local orthogonal manifolds $\Sigma$, also being in good
harmony with the tubular distance function $z_{tub}(\,.\,)$ defined earlier.
On any manifold $\Sigma_0^t(y)\cap U_0$ ($y\in D_0$) we define the smooth
field $\Cal X_{y,t}(y')$ ($y'\in \Sigma_0^t(y)\cap U_0$) of unit tangent
vectors of $\Sigma_0^t(y)\cap U_0$ as follows:
$$
\Cal X_{y,t}(y')=\frac{\Pi_{y,t,y'}\left((\delta\tilde q_0,\,\delta
\tilde v_0)\right)}
{\left\Vert\Pi_{y,t,y'}\left((\delta\tilde q_0,\,\delta
\tilde v_0)\right)\right\Vert},
\tag 4.13
$$
where $\Pi_{y,t,y'}$ denotes the orthogonal projection of
$\Bbb R^d\oplus\Bbb R^d$ onto the tangent space of $\Sigma_0^t(y)$ at the
point $y'\in\Sigma_0^t(y)\cap U_0$. Recall that
$(\delta\tilde q_0,\,\delta\tilde v_0)$ is the unit tangent vector of
$\bold M$ at the base point $x_0$ from (4.8) and (4.9). We also remind the
reader that $(\delta\tilde q_0,\,\delta\tilde v_0)$ points toward the side of
$J$ opposite to the side where the one-sided neighborhoods $U_\delta$ reside.
\medskip
\subheading{Note 4.14}
By the construction of $(\delta\tilde q_0,\delta\tilde v_0)$
in (4.6)--(4.8), if the threshold $c_3$ is big enough, then the vector
$(\delta\tilde q_0,\delta\tilde v_0)$ is close to the tangent space
$\Cal T\gamma^s(x_0)$ of the local stable manifold of $x_0$. On the other
hand, for large enough $t$ the tangent space of $\Sigma_0^t(y)\cap U_0$
at $y'$ makes a small angle with $\Cal T\gamma^s(x_0)$. All the necessary
upper estimations for the mentioned angles follow from the well known result
stating that the difference (in norm) between the second fundamental forms
of the $S^t$-images ($t>0$) of two local, convex orthogonal manifolds is at
most $1/t$, see, for instance, inequality (4) in [Ch(1982)]. These facts imply
that the vector in the numerator of (4.13) is actually very close to
$(\delta\tilde q_0,\delta\tilde v_0)$, in particular its magnitude is almost
one.
\medskip
For any $y\in D_0$ let $t_k=t_k(y)$ ($0c^*\sqrt{\delta_0}$ have a positive density amongst
the natural numbers.
\medskip
As far as the terminal point $\rho_{y,k}(h(y,k))$ of $\rho_{y,k}$ is
concerned, there are exactly three, mutually exclusive possiblities for this
point:
\medskip
(A) $\rho_{y,k}(h(y,k))\in J$ and this terminal point does not belong to any
forward singularity of order $\le k$,
\medskip
(B) $\rho_{y,k}(h(y,k))$ lies on a forward singularity of order $\le k$,
\medskip
(C) the terminal point $\rho_{y,k}(h(y,k))$ does not lie on any singularity
of order $\le k$ but lies on the part of the boundary
$\partial U_0$ of $U_0$ different from $J$.
\medskip
\subheading{Note 4.17} Under the canonical identification
$U_0\cong J\times[0,\,\delta_0)$ of $U_0$ via the geodesic lines
perpendicular to $J$, the above mentioned part of $\partial U_0$ (the "side"
of $U_0$) corresponds to $\partial J\times[0,\,\delta_0)$. Therefore, the set
of points with property (C) inside a layer $U_\delta$ ($\delta\le\delta_0$)
will have $\mu$-measure small ordo of $\delta$ (actually, of order
$\delta^2$), thus this set will be negligible in our asymptotic measure
estimations, as $\delta\to 0$. That is why in the future we will not be
dealing with any phase point with property (C).
\medskip
Should (B) occur for some value of $k$ ($k\ge 2$), the minimum of all such
integers $k$ will be denoted by $\overline{k}=\overline{k}(y)$. The exact
order of the forward singularity on which the terminal point
$\rho_{y,\overline{k}}\left(h(y,\overline{k})\right)$ lies is denoted by
$\overline{k}_1=\overline{k}_1(y)$. If (B) does not occur for any value of
$k$, then we take $\overline{k}(y)=\overline{k}_1(y)=\infty$.
We can assume that the manifold $J$ and its one-sided tubular neighborhood
$U_0=U_{\delta_0}$ are already so small that for any $y\in U_0$
no singularity of $S^{(0,\infty)}y$ can take place at the first
collision, so the indices $\overline{k}$ and $\overline{k}_1$ above are
automatically at least $2$. For our purposes the important index will be
$\overline{k}_1=\overline{k}_1(y)$ for phase points $y\in D_0$.
\medskip
\subheading{Note 4.18. Refinement of the construction} Instead of selecting a
single contracting unit vector $(\delta\tilde q_0,\,\delta\tilde v_0)$
in (4.8), we should do the following: Choose a compact set $K_0\subset B$ with
the following properties:
\medskip
(i) $\frac{\mu_1(K_0)}{\mu_1(J)}>1-10^{-6}$,
(ii) every point $x\in K_0$ has a non-singular forward orbit.
\medskip
Now the running point $x\in K_0$ will play the role of $x_0$ in the
construction of the contracting unit tangent vector
$u(x):=(\delta\tilde q_0,\,\delta\tilde v_0)\in\Cal T_x\bold M$ on the
left-hand-side of (4.8). For every $x\in K_0$ there is a small, open ball
neighborhood $B(x)$ of $x$ and a big threshold $c_3(x)>>1$ such that (4.9)
holds true for $u(x)$ and $c_3=c_3(x)$ for all $x\in K_0$.
By the continuity of the contraction/expansion factor, one can also achieve
that the contraction estimation $L_0^{-1}$ of (4.9) holds true not only for
$u(x)$, but also for any projected copy of it appearing in (4.13), provided
that $y'\in B(x)$, i. e. $y'$ is close enough to $x$.
Now select a finite subcovering $\bigcup_{i=1}^n B(x_i)$ of $K_0$,
and replace $J$ by $J_1=J\cap\bigcup_{i=1}^n B(x_i)$, $U_\delta$ by
$U'_\delta=U_\delta\cap\bigcup_{i=1}^n B(x_i)$
(for $\delta\le\delta_0$) and, finally,
choose the threshold $c_3$ to be the maximum of all thresholds $c_3(x_i)$
for $i=1,2,\dots,n$. In this way the assertion of
Corollary 4.20 will be indeed true.
We note that the new exceptional manifold $J_1$ is no longer so nicely "round
shaped" as $J$, but it is still pretty well behaved, being a
domain in $J$ with a piecewise smooth boundary.
The reason why we cannot switch completely to a round and much smaller
manifold $B(x)\cap J$ is that the measure $\mu_1(J)$ should be kept bounded
from below after having fixed $L_0$, see 4. in the Appendix.
In addition, it should be noted that, when constructing the vector field in
(4.13) and the curves $\rho_{y,t}$, an appropriate directing vector
$u(x_i)$ needs to be chosen for (4.13). To be definite and not arbitrary, a
convenient choice is the first index $i\in\{1,2,\dots,n\}$ for which
$y\in B(x_i)$. In that way the whole curve $\rho_{y,t}$ will stay in the
slightly enlarged ball $B'(x_i)$ with double the radius of $B(x_i)$, and one
can organize things so that the required contraction estimates of (4.9)
be still valid even in these enlarged balls.
In the future, a bit sloppily, $J_1$ will be denoted by $J$, and
$U'_\delta$ by $U_\delta$.
\medskip
\subheading{Note 4.19} When defining the returns of a forward orbit to
$U_\delta$, we used to say that "before every new return the orbit must first
leave the set $U_\delta$". Since the newly obtained $J$ is no longer round
shaped as it used to be, the above phrase is not satisfactory any longer.
Instead, one should say that the orbit leaves even the $\kappa$-neighborhood
of $U_\delta$, where $\kappa$ is two times the diameter of the original $J$.
This guarantees that not only the new $U_\delta$, but also the original
$U_\delta$ will be left by the orbit, so we indeed are dealing with a genuine
return. This note also applies to two more slight shrinkings of $J$ that will
take place later in the proof.
\medskip
As an immediate corollary of (4.9) and the above note, we get
\subheading{Corollary 4.20} For the given sets $J$, $U_0$, and the
large constant $L_0$ we can select the threshold $c_3>0$ large enough so
that for any point $y\in D_0$ any time $t$ with
$c_3\le t0$ is a constant,
independent of $L_0$, depending only on the (asymptotic) angles between the
curves $\rho_{y,\overline{k}(y)}$ and $J$.
\medskip
By further shrinking the exceptional manifold $J$ a little bit,
and by selecting a suitably
thin, one-sided neighborhood $U_1=U_{\delta_1}$ of $J$, we can achieve that
the open $2\delta_1$-neighborhood of $U_1$ (on the same side of $J$ as
$U_0$ and $U_1$) is a subset of $U_0$.
For a varying $\delta$, $0<\delta\le\delta_1$, we introduce the layer
$$
\aligned
\overline{U}_\delta=\Big\{&y\in(U_\delta \setminus U_{\delta/2})\cap
D_0\big|\;\exists\text{ a sequence }t_n\nearrow\infty \\
&\text{such that } S^{t_n}y\in(U_\delta \setminus U_{\delta/2})
\text{ for all }n\Big\}.
\endaligned
\tag 4.23
$$
Since almost every point of the layer
$(U_\delta \setminus U_{\delta/2})\cap D_0$ returns infinitely often to
this set and the asymptotic equation
$$
\mu\left((U_\delta \setminus U_{\delta/2})\cap D_0\right)\sim
\frac{\delta}{2}\mu_1(J)
$$
holds true, we get the asymptotic equation
$$
\mu\left(\overline{U}_\delta\right)
\sim\frac{\delta}{2}\mu_1(J).
\tag 4.24
$$
We will need the following subsets of $\overline{U}_\delta$:
$$
\aligned
\overline{U}_\delta(c_3)&=\left\{y\in\overline{U}_\delta\big|\;
t_{\overline{k}_1(y)-1}(y)\ge c_3\right\}, \\
\overline{U}_\delta(\infty)&=\left\{y\in\overline{U}_\delta\big|\;
\overline{k}_1(y)=\infty\right\}.
\endaligned
\tag 4.25
$$
Here $c_3$ is the constant from Corollary 4.20, the exact value of
which will be specified later, at the end of the proof of Main Lemma 4.5.
By selecting the pair of sets $(U_1,\,J)$ small enough, we can
assume that
$$
z_{tub}(y)>c_4\delta_1 \quad\forall y\in U_1.
\tag 4.26
$$
This inequality guarantees that the collision time $t_{\overline{k}_1(y)}(y)$
($y\in\overline{U}_\delta$) cannot be near any return time of $y$ to the
layer $(U_\delta\setminus U_{\delta/2})$, for $\delta\le\delta_1$,
provided that $y\in\overline{U}_\delta(c_3)$.
More precisely, the whole orbit segment
$S^{[-\tau(-z),\,\tau(z)]}z$ will be disjoint from $U_1$, where
$z=S^ty$, $t_{\overline{k}_1(y)-1}(y)0$) of
singularities can be smoothly foliated with local, concave orthogonal
manifolds. Thus, the $\delta$-neighborhood of this singularity set inside
$\overline{U}_\delta$ clearly has $\mu$-measure small ordo of $\delta$,
actually, of order $\le\text{const}\cdot\delta^2$. \qed
\medskip
\subheading{Lemma 4.28} For any point $y\in\overline{U}_\delta(\infty)$
the curves $\rho_{y,k}(s)$ ($0\le s\le h(y,k)$) have a $C^2$-limiting curve
$\rho_{y,\infty}(s)$ ($0\le s\le h(y,\infty)$), with
$h(y,\,k)\to h(y,\,\infty)$, as $k\to\infty$.
\medskip
\subheading{Proof} Besides the concave, local orthogonal manifolds
$\Sigma_0^k(y)=\Sigma_0^{t_k^*}(y)$ of (4.12) (where
$t_k^*=t_k^*(y)=\frac{1}{2}\left(t_{k-1}(y)+t_k(y)\right)$), let us also
consider another type of concave, local orthogonal manifolds defined by
the formula
$$
\tilde\Sigma_0^k(y)=
\tilde\Sigma_0^{t_k^*}(y)=SC_y\left(S^{-t_k^*}\left(SC_{y_{t_k^*}}\left\{y'\in
\bold M\big|\; q(y')=q(y_{t_k^*})\right\}\right)\right),
\tag 4.29
$$
the so called "candle manifolds", containing the phase point
$y\in\overline{U}_\delta(\infty)$ in their interior. It was proved in \S3
of [Ch(1982)] that the second fundamental forms
$B\left(\Sigma_0^k(y),\,y\right)\le 0$ are monotone
non-increasing in $k$, while the second fundamental forms
$B\left(\tilde\Sigma_0^k(y),\,y\right)<0$ are monotone
increasing in $k$, so that
$$
B\left(\tilde\Sigma_0^k(y),\,y\right)****c_3$ for which
$S^ty\in\overline{U}_\delta\subset\left(U_\delta\setminus
U_{\delta/2}\right)\cap D_0$. The distance $\text{dist}(S^ty,\,J)$ between
$S^ty$ and $J$ is bigger than $\delta/2$. According to the contraction
result 4.20, if the contraction factor $L_0^{-1}$ is chosen small enough, the
distance between $S^{t}\left(\Pi(y)\right)$ and $J$ stays bigger than
$\delta/4$, so $S^{t}\left(\Pi(y)\right)\in U_0\setminus J$ will be true.
This means, however, that the forward orbit of $\Pi(y)$ is sufficient,
according to (4)/(ii) of Definition 4.1. \qed
\medskip
\subheading{Corollary 4.31} $\mu\left(\overline{U}_\delta(\infty)\right)=0$.
\medskip
\subheading{Proof} By our indirect assumption $A=\emptyset$, so the previous
lemma says that
$$
\Pi(y)\in\bigcup_{t>0} S^{-t}\left(\Cal S\Cal R^-\right)
$$
for all $y\in\overline{U}_\delta(\infty)$. Due to the transversality result
4.3 above, the singularity set
$\bigcup_{t>0} S^{-t}\left(\Cal S\Cal R^-\right)$ is a countable union of
smooth, proper submanifolds of $J$. Since the curves $\rho_{y,\infty}$ belong
to (at least) the $C^2$ smoothness class, we get that
$\overline{U}_\delta(\infty)$ is a countable union of proper, $C^2$-smooth
submanifolds of $U_\delta$, hence
$\mu\left(\overline{U}_\delta(\infty)\right)=0$. \qed
\medskip
Next we need a useful upper estimation for the $\mu$-measure of the set
$\overline{U}_\delta(c_3)\setminus\overline{U}_\delta(\infty)$ as
$\delta\to 0$. We will classify the points
$y\in\overline{U}_\delta(c_3)\setminus\overline{U}_\delta(\infty)$
according to whether $S^ty$ returns to the layer
$\left(U_\delta\setminus U_{\delta/2}\right)\cap D_0$ (after first
leaving it, of course) before the time $t_{\overline{k}_1(y)-1}(y)$ or not.
Thus, we define the sets
$$
\aligned
E_\delta(c_3)=&\big\{y\in\overline{U}_\delta(c_3)\setminus
\overline{U}_\delta(\infty)\big|\; \exists\; 01-10^{-6},
$$
and no point of $K_1$ ever returns to $J$. For each point $x\in K_1$ the
distance between the orbit segment $S^{[a_0, c_3]}x$ and $J$ is at least
$\epsilon(x)>0$. Here $a_0$ is needed to guarantee that we certainly drop
the initial part of the orbit, which still stays near $J$, and $c_3$ was chosen
earlier. By the non-singularity of the orbit segment $S^{[a_0, c_3]}x$ and
by continuity, the point $x\in K_1$ has an open ball neighborhood $B(x)$
of radius $r(x)>0$ such that for every $y\in B(x)$ the orbit segment
$S^{[a_0, c_3]}y$ is non-singular and stays away from $J$ by at least
$\epsilon(x)/2$. Choose a finite covering
$\bigcup_{i=1}^n B(x_i)\supset K_1$ of $K_1$, replace $J$ and $U_\delta$ by
their intersections with the above union (the same way as it was done in
Note 4.18), and fix threshold value of $\delta_0$ so that
$$
\delta_0<\frac{1}{2}\min\{\epsilon(x_i)|\, i=1,2,\dots,n\}.
$$
In the future we again keep the old notations $J$ and $U_\delta$ for these
intersections.
In this way we achieve that the following statement is true:
$$
\cases
&\text{Any return time }t_2\text{ of any point }y\in
\left(U_\delta\setminus U_{\delta/2}\right)\cap D_0\text{ to} \\
&\left(U_\delta\setminus U_{\delta/2}\right)\cap D_0
\text{ is always greater than }c_3\text{ for } 0<\delta\le\delta_1.
\endcases
\tag 4.33
$$
For any phase point $y\in E_\delta(c_3)$ we define the first return time
$\overline{t}_2=\overline{t}_2(y)$ as the infimum of all the return times
$t_2$ of $y$ featuring (4.32). By using this definition of
$\overline{t}_2(y)$, formulas (4.32)--(4.33), and the contraction result
4.20, we easily get
\medskip
\subheading{Lemma 4.34} If the contraction coefficient $L_0^{-1}$ in 4.20 is
chosen suitably small, then for any point $y\in E_\delta(c_3)$ the
projected point
$$
\Pi(y):=\rho_{y,\overline{t}_2(y)}\left(h(y,\overline{t}_2(y))\right)\in J
\tag 4.35
$$
is either a forward singular point of $J$, or $\Pi(y)$ belongs to
the set $A$ of regular sufficient points of $J$, defined in 4.4.
\medskip
\subheading{Proof} Since $\overline{t}_2(y)0$ is the geometric constant (also denoted by $c_2$) in Lemma 2 of
[S-Ch(1987)] or in Lemma 4.10 of [K-S-Sz(1990)-I], $c_4$ is the constant in
(4.22) above, and $\nu$ is the natural $T$-invariant measure on
$\partial\bold M$ that can be obtained by projecting the Liouville measure
$\mu$ onto $\partial\bold M$ along the billiard flow.
\medskip
\subheading{Proof} Let $y\in F_\delta(c_3)$. From the inequality
$t_{\overline{k}_1(y)-1}(y)\ge c_3$ and from Corollary 4.21 we conclude that
$z_{tub}\left(\Pi(y)\right)0$ is small
enough.
\medskip
\subheading{Proof} The relation $\Pi(y_1)=\Pi(y_2)$ implies that $y_1$ and
$y_2$ belong to the same orbit, so we can assume, for example, that
$y_2=S^ay_1$ with some $a>0$. We need to prove that
$S^{[0,a]}y_1\subset U_0$. Assume the opposite, i. e. that there is a number
$t_1$, $0a$, see (4.37).
The obtained inequality
$t_{\overline{k}_1(y_1)-1}(y_1)>a\ge\overline{t}_2(y)$, however,
contradicts to the definition of the set $F_\delta(c_3)$, to which $y_1$
belongs as an element, see (4.32). The upper estimate $1.1\text{diam}(J)$
for the length of $S$ is an immediate corollary of the containment
$S\subset U_0$. \qed
\medskip
As a direct consequence of lemmas 4.38 and 4.39, we obtain
\medskip
\subheading{Corollary 4.40} For all small enough $\delta>0$ the inequality
$$
\mu\left(F_\delta(c_3)\right)\le 1.1c_2c_4L_0^{-1}\delta\text{diam}(J)
$$
holds true.
\bigskip
\subheading{Finishing the Indirect Proof of Main Lemma 4.5}
\medskip
It follows immediately from Lemma 4.27 and corollaries 4.31, 4.36, and 4.40
that
$$
\mu\left(\overline{U}_\delta\right)\le 1.2c_2c_4\text{diam}(J)L_0^{-1}\delta
$$
for all small enough $\delta>0$. This fact, however, contradicts to (4.24)
if $L_0^{-1}$ is selected so small that
$$
1.2c_2c_4\text{diam}(J)L_0^{-1}<\frac{1}{4}\mu_1(J^*),
$$
where $J^*$ stands for the original exceptional manifold before the three
slight shrinkings in the style of Note 4.18. Clearly,
$\mu_1(J)>(1-10^{-5})\mu_1(J^*)$. The obtained contradiction finishes the
indirect proof of Main Lemma 4.5. \qed
\bigskip \bigskip
\heading
\S5. Proof of Ergodicity \\
The Induction on $N$
\endheading
\bigskip
By using several results of Sinai [Sin(1970)], Chernov-Sinai [S-Ch(1987)], and
Kr\'amli-Sim\'anyi-Sz\'asz, in this section we finally prove the ergodicity
(hence also the Bernoulli property, see Chernov-Haskell [C-H(1996)] or
Ornstein-Weiss [O-W(1998)]) for every hard ball system $(\bold
M,\,\{S^t\},\,\mu)$, under the assumption of the Ansatz for the considered
hard ball system $\flow$ and for all of its subsystems, by carrying out an
induction on the number $N$ ($\ge 2$) of interacting balls.
The base of the induction (i. e. the ergodicity of any two-ball system on a
flat torus) was proved in [Sin(1970)] and [S-Ch(1987)].
Assume now that $(\bold M,\,\{S^t\},\,\mu)$ is a given system of $N$ ($\ge 3$)
hard spheres with masses $m_1,m_2,\dots,m_N$ and radius $r>0$ on the flat unit
torus $\Bbb T^\nu=\Bbb R^\nu/\Bbb Z^\nu$ ($\nu\ge2$), as defined in \S2.
Assume further that the ergodicity of every such system is already proved to
be true for any number of balls $N'$ with $2\le N'0$ so that $S^Tx_0\not\in\partial\bold M$, and the
symbolic collision sequence $\Sigma_0=\Sigma\left(S^{[0,T]}x_0\right)$ is
combinatorially rich in the sense of Definition 3.28 of [Sim(2002)]. By
further shrinking $J$, if necessary, we can assume that
$S^T(J)\cap\partial\bold M=\emptyset$ and $S^T$ is smooth on $J$. Choose a
thin, tubular neighborhood $\tilde{U}_0$ of $J$ in $\bold M$ in such a way
that $S^T$ be still smooth across $\tilde{U}_0$, and define the set
$$
NS\left(\tilde{U}_0,\,\Sigma_0\right)=\left\{x\in\tilde{U}_0\big|\;\text{dim}
\Cal N_0\left(S^{[0,T]}x\right)>1\right\}
\tag 5.1
$$
of not $\Sigma_0$-sufficient phase points in $\tilde{U}_0$.
Clearly, $J\subset NS\left(\tilde{U}_0,\,\Sigma_0\right)$. We can
assume that the selected (generic) base point $x_0\in J$ belongs to the smooth
part of the closed algebraic set $NS\left(\tilde{U}_0,\,\Sigma_0\right)$.
This guarantees that actually $J=NS\left(\tilde{U}_0,\,\Sigma_0\right)$, as
long as the manifold $J$ and its tubular neighborhood are selected small
enough, thus achieving property (4) of 4.1.
\medskip
\subheading{Proof of why property (6) of Definition 4.1 can be assumed}
\medskip
We recall that $J$ is a codimension-one, smooth manifold of non-sufficient
phase points separating two open ergodic components, as described in (0)--(3)
at the end of \S3 of [Sim(2003)].
Let $P$ be the subset of $J$ containing all points with non-singular forward
orbit and recurring to $J$ infinitely many times.
\medskip
\subheading{Lemma 5.2} $\mu_1(P)=0$.
\medskip
\subheading{Proof} Assume that $\mu_1(P)>0$. Take a suitable Poincar\'e
section to make the time discrete, and consider the on-to-one first return map
$T:\;P\to P$ of $P$. According to the measure expansion theorem for
hypersurfaces $J$ (with negative infinitesimal Lyapunov function $Q(n)$ for
their normal field $n$), proved in [Ch-Sim(2006)], the measure
$\mu_1\left(T(P)\right)$ is strictly larger than $\mu_1(P)$, though
$T(P)\subset P$. The obtained contradiction proves the lemma. \qed
\medskip
Next, we claim that the above lemma is enough for our purposes to prove (6) of
4.1. Indeed, the set $W\subset J$ consisting of all points $x\in J$ never
again returning to $J$ (after leaving it first, of course) has positive
$\mu_1$-measure by Lemma 5.2. Select a Lebesgue density base point $x_0\in W$
for $W$ with a non-singular forward orbit, and shrink $J$ at the very
beginning to such a small size around $x_0$ that the relative measure of $W$
in $J$ be bigger than $1-10^{-8}$.
\medskip
Finally, Main Lemma 4.5 asserts that $A\ne\emptyset$,
contradicting to our earlier statement that no point of $J$ is
sufficient. The obtained contradiction completes the inductive step of the
proof of the Theorem. \qed
\bigskip
\heading
Appendix. The Constants of \S3--4
\endheading
\bigskip
In order to make the reading of sections 3--4 easier, here we briefly describe
the hierarchy of the constants used in those sections.
\medskip
1. The geometric constant $c_0>0$ of Proposition 3.5 is a lower estimation for
the "curvature" $\langle\delta q_0,\,\delta v_0\rangle/||\delta q_0||^2$
of an expanding tangent vector $(\delta q_0,\,\delta v_0)$.
\medskip
2. The geometric constant $-c_1<0$ provides an upper estimation for the
infinitesimal Lyapunov function $Q(n)$ of $J$ in (5) of Definition 4.1. It
cannot be freely chosen in the proof of Main Lemma 4.5.
\medskip
3. The constant $c_2>0$ is present in the upper measure estimation of Lemma 2
of [S-Ch(1987)], or Lemma 4.10 in [K-S-Sz(1990)-I]. It cannot be changed in
the course of the proof of Main Lemma 4.5.
\medskip
4. The contraction coefficient $0>1$ large
enough (see Corollary 4.20), after having fixed $U_0$, $\delta_0$, and $J$.
The phrase "suitably small" for $L_0^{-1}$ means that the inequality
$$
L_0^{-1}<\frac{0.25\mu_1(J^*)}{1.2c_2c_4\text{diam}(J)}
$$
should be true, see the end of \S4.
\medskip
5. The geometric constant $c_4>0$ of (4.22) bridges the gap between two
distances: the distance $\text{dist}(y,J)$ between a point $y\in U_\delta$ and
$J$, and the arc length $l_q\left(\rho_{y,\overline{k}(y)}\right)$. It cannot
be freely chosen during the proof of Main Lemma 4.5.
\bigskip
\subheading{Acknowledgement} The author expresses his sincere gratitude
to N. I. Chernov for his numerous, very valuable questions, remarks, and
suggestions.
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**