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\noindent
March 21, 1997
\bigskip \bigskip
\centerline{Ergodicity of Hard Spheres in a Box}
\vskip1cm
\centerline{{N\'andor Sim\'anyi}
\footnote{Research supported by the Hungarian National Foundation for
Scientific Research, grants OTKA-7275 and OTKA-16425.}}
\centerline{Bolyai Institute of J\'ozsef Attila University}
\centerline{1 Aradi V\'ertanuk tere, Szeged, H-6720 Hungary}
\centerline{E-mail: simanyi\@math.u-szeged.hu}
\bigskip \bigskip
\hbox{\centerline{\vbox{\hsize 8cm {Abstract.}
We prove that the system of two hard balls in a $\nu$-dimensional
($\nu\ge 2$) rectangular box is ergodic and, therefore, actually it is a
Bernoulli flow.}}}
\bigskip \bigskip
\centerline{1. Introduction}
\bigskip \bigskip
\heading
The Model and the Theorem
\endheading
\bigskip
Let us consider the billiard system of two hard balls with unit mass and
radius $r$ ($02$
took fourteen years, and was done by Chernov and Sinai in [21].
Although the model of two hard balls in $\Bbb T^\nu$ is already
rather involved technically, it is still a so called strictly dispersive
billiard system, i. e. such that the smooth components of the boundary
$\partial\bold Q$ of the configuration space are strictly convex from
inside $\bold Q$. The billiard systems of more than two hard spheres in
$\Bbb T^\nu$ are no longer strictly dispersive, but just dispersive
(strict convexity of the smooth components of $\partial\bold Q$
is lost, merely convexity persists!), and this circumstance causes a lot
of additional technical troubles in their study. In the series of my joint
papers with A. Kr\'amli and D. Sz\'asz [9--12]
we developed several new methods, and proved the ergodicity of
more and more complicated semi-dispersive billiards culminating in the proof
of the ergodicity of four billiard balls in the torus $\Bbb T^\nu$
($\nu\ge 3$), [12]. Then, in 1992, Bunimovich, Liverani,
Pellegrinotti and Sukhov [1] were able to prove the ergodicity for
some systems with an arbirarily large number of hard balls. The shortcoming
of their model, however, is that, on one hand, they restrict the types of all
feasible ball-to-ball collisions, on the other hand, they introduce some extra
scattering effect with the collisions at the strictly convex wall of the
container. The only result with an arbirarily large number of spheres in a
flat torus $\Bbb T^\nu$ was achieved in [15--16], where I managed to
prove the ergodicity (actually, the K-mixing property) of $N$ hard balls in
$\Bbb T^\nu$, provided that $N\le\nu$. The annoying shortcoming of that result
is that the larger the number of balls $N$ is, larger and larger dimension
$\nu$ of the ambient container is required by the method of the proof.
Finally, in our latest joint venture with D. Sz\'asz [17] we prevailed
over the difficulty caused by the low value of the dimension $\nu$ by
developing a brand new algebraic approach for the study of hard ball systems.
That result, however, only establishes hyperbolicity (nonzero Lyapunov
exponents almost everywhere) for $N$ balls in $\Bbb T^\nu$. The ergodicity is
a bit longer shot...
None of the above results took up the problem of handling hard balls in
physically more realistic containers, e. g. rectangular boxes. The extra
technical hardship in their investigation is caused by the loss of the
total momentum and center of mass. This amounts to the increase in the
dimension of the configuration (phase) space without any additional
scattering effect as a compensation. The problem of proving ergodicity for
$N$ hard spheres in a $\nu$-dimensional rectangular box is so difficult that
we were only able to achieve that goal in the case $N=2$.
\bigskip
\heading
The Strategy of the Proof
\endheading
\bigskip
After reviewing the necessary technical skills in Section 2, in the subsequent
section we introduce the concept of combinatorial richness of the symbolic
collision structure of a trajectory (segment), and then we prove that ---
apart from some codimension-two, smooth submanifolds of the phase space ---
such a combinatorial richness implies the sufficiency (hyperbolicity) of the
trajectory.
The proof of the theorem goes on by an induction on the number
$k=0,1,\dots ,\nu$. Section 4 uses the inductive hypothesis (the statement
of the theorem for $k-1$) and proves that the set of combinatorially
non-rich phase points is {\it slim}, i. e. it can be covered by a countable
family of closed, zero measure sets with codimension at least two.
Section 5 contains the inductive proof of the theorem based upon the preceding
two sections and the celebrated Theorem on Local Ergodicity for
semi-dispersive billiards by Chernov and Sinai, [21].
The closing Appendix is a brief overview of a special orthogonal cylindric
billiard which emerged in Section 4.
\bigskip \bigskip
\centerline{2. Prerequisites}
\bigskip \bigskip
\centerline{Semi-dispersive Billiards}
\bigskip
A billiard is a dynamical system describing the
motion of a point particle in a connected, compact domain
$\bold Q\subset{\Bbb R}^d$
or $\bold Q\subset{\Bbb T}^d=\Bbb R^d/\Bbb Z^d$, $d\ge 2$, with a piecewise
$C^2$-smooth boundary. Inside $\bold Q$ the motion is uniform, whereas the
reflection at the boundary $\partial\bold Q$ is elastic (the angle of
reflection equals the angle of incidence). Since the absolute value
of the velocity is a first integral of the motion, the phase space of our
system can be identified with the unit tangent bundle over $\bold Q$.
Namely, the configuration space is $\bold Q$, while the phase space is
$\bold M=\bold Q\times\Bbb S^{d-1}$,
where $\Bbb S^{d-1}$ is the unit $d-1$-sphere.
In other words, every phase point $x$ is of the form
$(q,v)$ where $q\in\bold Q$ and $v\in\Bbb S^{d-1}$ is a tangent vector
at the footpoint $q$. The natural
projections $\pi:\,\bold M\to\bold Q$ and $p:\,\bold M\to\Bbb S^{d-1}$
are defined by $\pi(q,v)=q$ and $p(q,v)=v$, respectively.
Suppose that $\partial\bold Q=\cup_1^k\partial\bold Q_i$, where
$\partial\bold Q_i$ are the smooth components of the boundary. Denote
$\partial\bold M=\partial\bold Q\times\Bbb S^{d-1}$,
and let $n(q)$ be the unit
normal vector of the boundary component $\partial\bold Q_i$ at
$q\in\partial\bold Q_i$ directed inwards $\bold Q$.
The flow $\left\{S^t:\, t\in{\Bbb R}\right\}$ is determined for the subset
$\bold M'\subset\bold M$ of phase points whose trajectories never cross the
intersections of the smooth pieces of $\partial\bold Q$ and do not contain
an infinite number of reflections in a finite time interval. If $\mu$
denotes the (normalized) Liouville measure on $\bold M$, i.e.
$d\mu(q,v)=\text{const}\cdot dq\cdot dv$,
where $dq$ and $dv$ are the differentials of the
Lebesgue measures on $\bold Q$ and on $\Bbb S^{d-1}$,
respectively, then under certain conditions $\mu(\bold M')=1$ and $\mu$
is invariant [8]. The interior of the phase space $\bold M$
can be endowed with the natural Riemannian metric.
The dynamical system $\left(\bold M,\{S^t\},\mu\right)$ is called a
{\it billiard}.
Notice, that $\left(\bold M,\{S^t\},\mu\right)$ is neither everywhere
defined nor smooth.
The main object of the present paper is a particularly interesting
class of billiards: the {\it semi-dispersive billiards} where, for
every $q\in\partial\bold Q$ the second fundamental form $K(q)$ of the boundary
is positive semi-definite.
If, moreover, for every $q\in \partial\bold Q$ the second fundamental form
$K(q)$ is positive definite,
then the billiard is called a {\it dispersive billiard}.
\medskip
As it is pointed out in previous works on billiards, the dynamics can
only be defined for trajectories where the moments of collisions do not
accumulate in any finite time interval (cf. Condition 2.1 of
[10]). An important consequence of Theorem 5.3 of [23] is
that -- for semi-dispersing billiards -- there are {\it no trajectories
at all with a finite accumulation point of collision moments},
see also [6].
\bigskip
\centerline{Convex Orthogonal Manifolds}
\bigskip
In the construction of invariant manifolds a crucial role is played
by the time evolution equation for the second fundamental form of
codimension-one submanifolds in $\bold Q$ orthogonal to the velocity
component $p(x)$ of a phase point $x$.
Let $x=(q,v)\in\bold M\setminus\partial\bold M$, and consider a
$C^2$-smooth, codimension-one submanifold
$\tilde{\Cal O}\subset\bold Q\setminus\partial\bold Q$
such that $q\in\tilde{\Cal O}$ and $v=p(x)$ is a
unit normal vector to $\tilde {\Cal O}$ at $q$. Denote by $\Cal O$
the normal section of the unit tangent bundle of $\bold Q$ restricted to
$\tilde {\Cal O}$. (The manifold $\Cal O$ is uniqely defined by the
orientation $(q,v)\in\Cal O$.) We call
$\Cal O$ a {\it local orthogonal manifold} with support $\tilde\Cal O$.
Recall that {\it the second fundamental form} $B_{\Cal O}(x)$ of
$\Cal O$ (or $\tilde{\Cal O}$) at $x$ is defined through
$$
v(q+\delta q)-v(q)=B_{\Cal O}(x)\cdot\delta q+
o\left(\Vert\delta q\Vert\right),
$$
and it is a self-adjoint operator acting in the
$(d-1)$-dimensional tangent hyperplane of $\tilde{\Cal O}$ at $q$.
\medskip
A local orthogonal manifold $\Cal O$ is called {\it convex} if
$B_{\Cal O}(y)\ge 0$ for every $y\in\Cal O$.
Recall that the common tangent space $\Cal T_q\bold Q$ of the
parallelizable configuration
space $\bold Q\subset\Bbb R^d$ (or $\bold Q\subset\Bbb T^d$) at the point
$q\in\text{int}\bold Q$ is the Euclidean space $\Bbb R^d$.
\bigskip
\centerline{Neutral Vectors, Advance and Sufficiency}
\bigskip
Consider a {\it non-singular} trajectory segment $S^{[a,b]}x$
in a semi-dispersive billiard.
Suppose that $a$ and $b$ are {\it not moments of collision}.
\medskip
\proclaim{Definition 2.1} The neutral space $\Cal N_0(S^{[a,b]}x)$
of the trajectory segment $S^{[a,b]}x$ at time zero ($a<0**0)
\text{ such that }\forall\alpha\in(-\delta,\, \delta) \\
& p\left(S^a\left(\pi(x)+\alpha w,\, p(x)\right)\right)=p(S^ax)\text{ and} \\
& p\left(S^b\left(\pi(x)+\alpha w,\, p(x)\right)\right)=p(S^bx)\big\}.
\endaligned
$$
\endproclaim
\medskip
It is known (see (3) in Section 3 of [21]) that
$\Cal N_0(S^{[a,b]}x)$ is a linear subspace of $\Bbb R^d$ indeed, and
$V(x)\in\Cal N_0(S^{[a,b]}x)$. The neutral space $\Cal N_t(S^{[a,b]}x)$
of the segment $S^{[a,b]}x$ at time $t\in [a,b]$ is defined as follows:
$$
\Cal N_t(S^{[a,b]}x)=\Cal N_0\left(S^{[a-t,b-t]}(S^tx)\right).
$$
It is clear that the neutral space $\Cal N_t(S^{[a,b]}x)$ can be canonically
identified with $\Cal N_0(S^{[a,b]}x)$ by the usual identification of the
tangent spaces of $\bold Q$ along the trajectory $S^{(-\infty,\infty)}x$
(see, for instance, Section 2 of [10]).
\medskip
Our next definition is that of the {\it advance}. Consider a
non-singular orbit segment $S^{[a,b]}x$ with a certain collision $\sigma$
taking place at time $t=t(x,\, \sigma)$. For $x=(q,v)\in\bold M$ and
$w\in\Bbb R^d$, $\Vert w\Vert$ sufficiently small, introduce the notation
$T_w(q,v):=(q+w,v)$.
\medskip
\proclaim{Definition 2.2} For any collision $\sigma$ of $S^{[a,b]}x$
and for any $t\in[a,b]$, the advance
$$
\alpha_\sigma :\, \Cal N_t(S^{[a,b]}x)\rightarrow\Bbb R
$$
is the unique linear extension of the linear functional
defined in a sufficiently small neighborhood of the origin of
$\Cal N_t(S^{[a,b]}x)$ in the following way:
$$
\alpha_\sigma (w):=t(x,\sigma)-t(S^{-t}T_w S^tx,\sigma).
$$
\endproclaim
\medskip
The overall important feature of the neutral space
$\Cal N_0(S^{[a,b]}x)$ is the following one:
For every given ball-to-ball collision $\sigma$ of the trajectory
$S^{[a,b]}x$, for every fixed time $t0$)}
\bigskip \bigskip
In this section we will be studying the neutral spaces and sufficiency of
{\it non-singular} trajectory segments $\omega=S^{[a,b]}x_0$, where $a$ and
$b$ are not moments of collision. We are interested in answering the following
fundamental question: How does the neutral space $\Cal N(\omega)$ depend on
the symbolic collision sequence $\Sigma(\omega)=\Sigma$ of the given orbit
segment $\omega=S^{[a,b]}x_0$, and what kind of combinatorial {\it richness}
of $\Sigma$ guarantees sufficiency, i. e. that $\text{dim}\Cal N(\omega)=1$?
\bigskip
\centerline{The Symbolic Sequence $\Sigma(\omega)$}
\bigskip
Let us symbolically denote the really dispersive (i. e. ball-to-ball)
collisions $\{1,2\}$ of the orbit segment $\omega$ by
$\sigma_0,\sigma_1,\dots ,\sigma_n$ just as they follow each other in time,
and denote by $a0$. Denote by $U_0$ the $\epsilon_0$-neighborhood of the base
point $x_0$ in $J_1$:
$$
U_0=\left\{x\in J_1|\, d(x_0,x)<\epsilon_0 \right\}.
$$
Besides $\epsilon_0$, select another, small positive number
$\delta_0$ such that for every $x\in U_0$ the trajectory segment
$S^{[0,\delta_0]}x$ does not have a collision, not even a non-scattering
ball-to-wall collision.
We will foliate $U_0$ by {\it convex, local orthogonal manifolds}
(see Section 2) as follows: We introduce the equivalence relation $\sim$
in $U_0$ by the formula
$$
x\sim y\Leftrightarrow (\exists \, \lambda\in\Bbb R) \quad q_1^0(x)-q_1^0(y)=
q_2^0(y)-q_2^0(x)=\lambda\cdot e_{j_1}
\tag 3.11
$$
for $x,y\in U_0$, where $e_{j_1}$ is the standard unit vector in the positive
direction of the $j_1$-st coordinate axis. It is easy to see that, indeed,
the equivalence classes $C(x)\subset U_0$ of $\sim$ are
$(\nu+k-1)$--dimensional, smooth, connected submanifolds of $U_0$.
Furthermore, the positive images $S^t\left(C(x)\right)$ ($00$ and $j_1\in Z_l$.
Such an index $l_0$ exists by the assumption of the lemma.
By using an induction on the integers $l=0,1,\dots ,l_0-1$, easy calculation
proves that for every non-negative integer $l0$, and $x_0$ is also a smooth point of the
corresponding singularity set $\Cal S\ni x_0$. We claim that $\hat J_1$
and $\Cal S$ intersect each other transversally at $x_0$.
\endproclaim
\subheading{Proof} Following the proof of the previous lemma, we again
consider the foliation $U_0\cap \hat J_1=\bigcup_{\alpha\in A}C(x_\alpha)$,
based on the equivalence relation (3.11), where $A$ is a suitable index set.
As we have seen in the previous proof, for every small $t>0$ the set
$S^t\left(C(x_\alpha)\right)$ is, indeed, a local, convex, orthogonal
manifold. Sublemma 4.2 of [10] then yields that the manifold
$C(x_0)\ni x_0$ is transversal to $\Cal S$ at the base point $x_0$.
Hence the lemma follows. \qed
\bigskip \bigskip
\centerline{4. Richness Is Abundant}
\bigskip \bigskip
In this section we will be investigating {\it non-singular} trajectories
$$
S^{(-\infty,\infty)}x_0=\left\{x_t=(q_1^t,q_2^t;\, \dot q_1^t,\dot q_2^t)
|\, t\in\Bbb R\right\}
$$
of the standard $(\nu,k,r)$-flow with $k>0$. (Sometimes we supress the
time derivatives $\dot q_i^t$, and just simply write $x_t=(q_1^t,q_2^t)$.)
The singular orbits will be taken care of in Section 5.
First of all, we note that, according to the theorems by Vaserstein
[23] and Galperin [6], the moments of collisions on the
trajectory $S^{(-\infty,\infty)}x_0$ can not accumulate at a finite point,
i. e. every bounded time interval only contains finitely many collisions.
The next thing we want to make sure is that there are infinitely many
scattering (i. e. ``ball-to-ball'') collisions both in
$S^{[0,\infty)}x_0$ and in $S^{(-\infty,0]}x_0$.
\medskip
\proclaim{Lemma 4.1} There exist finitely many nonzero vectors
$z_1,\dots ,z_l$ in $\Bbb R^{\nu-k}$ with the following property:
If a non-singular trajectory $S^{(-\infty,\infty)}x_0$ has no ball-to-ball
collision on a time interval $(t_0,\infty)$, then either
(1) $x_0$ belongs to some slim exceptional subset of the phase space
$\bold M$,
\noindent
or
(2) the projection $P_{\overline{\Cal A}}\left(v_1^{t_0}-v_2^{t_0}
\right)\in\Bbb R^{\nu-k}$ of the relative velocity (measured at $t_0$)
into the component $\Bbb R^{\nu-k}$ of the velocity space
$\Bbb R^\nu=\Bbb R^k\oplus\Bbb R^{\nu-k}=\Bbb R^{\Cal A}\oplus
\Bbb R^{\overline{\Cal A}}$ is perpendicular to some vector $z_i$.
In either case the entire trajectory $S^{(-\infty,\infty)}x_0$
does not contain a single ball-to-ball collision.
\endproclaim
\medskip
\subheading{Remark} In the case $k=\nu$ we have that $l=0$ and the
second possibility (2) will not occur.
\medskip
\subheading{Proof} Assume that $t_0=0$, i. e. the positive orbit
$$
S^{[0,\infty)}x_0=\left\{x_t=(q_1^t,q_2^t)|\, t\ge 0\right\}
$$
does not have any ball-to-ball collision.
Then there is a quite standard method of
``unfolding'' this positive orbit by reflecting the container
$\bold C=[0,1]^k\times\Bbb T^{\nu-k}$ across its boundary hyperplanes
$(q)_j=0$, $(q)_j=1$, $j=1,\dots ,k$. Namely, we select first an arbitrary
Euclidean lifting $\tilde q_1^0,\, \tilde q_2^0\in\Bbb R^\nu$ of the initial
positions $q_1^0,\, q_2^0\in[0,1]^k\times\Bbb T^{\nu-k}$ by not altering the
first $k$ coordinates, that is, the non-periodic ones. Then we extend this
lifting to $t\ge 0$ in a linear manner:
$\tilde q_i^t:=\tilde q_i^0+tv_i^0$, $t\ge 0$, $i=1,2$. (We note that
$(\tilde q_1^t,\, \tilde q_2^t)$ is {\it not} a lifting of the original
orbit.) Finally, we project the linear orbit
$\left\{(\tilde q_1^t,\, \tilde q_2^t)|\, t\ge 0 \right\}$ into the torus
$$
\left(\Bbb R^k/2\cdot\Bbb Z^k\right)\times\Bbb T^{\nu-k}=
\Bbb R^\nu/\left[\left(2\cdot\Bbb Z^k\right)\times\Bbb Z^{\nu-k}\right]
$$
by using the natural projection
$$
\Bbb R^\nu\longrightarrow
\Bbb R^\nu/\left[\left(2\cdot\Bbb Z^k\right)\times\Bbb Z^{\nu-k}\right],
$$
and obtain the positive trajectory
$\left\{(\hat q_1^t,\, \hat q_2^t)|\, t\ge 0\right\}$. It is obvious that the
finally obtained orbit is independent of the initial selection of the lifting.
Let us define the canonical ``folding map''
$$
\Phi=(\phi,\dots ,\phi;\, \text{id},\dots ,\text{id}):\,
\left(\Bbb R^k/2\cdot\Bbb Z^k\right)\times\Bbb T^{\nu-k}\rightarrow
[0,1]^k\times\Bbb T^{\nu-k}=\bold C
$$
in such a way that the same 2-periodic rooftop function
$\phi:\, \Bbb R/2\cdot\Bbb Z\rightarrow [0,1]$,
$\phi(x):=d(x,\, 2\cdot\Bbb Z)$, acts on the first $k$
components, while the identity function acts on the remaining coordinates.
One easily sees that
$\Phi\left((\hat q_1^t,\, \hat q_2^t)\right)=(q_1^t,\, q_2^t)=x_t$.
The fact that the original positive orbit $S^{[0,\infty)}x_0$ has no
ball-to-ball collision immediately implies these assertions:
(i) $d(\hat q_1^t,\, \hat q_2^t)>2r$ for all $t\ge 0$;
(ii) $d\left(\hat q_1^t,\, R_{\Cal A}\hat q_2^t\right)>2r$ for all $t\ge 0$,
\noindent
where $d(.,\, .)$ is the usual Euclidean distance in
$\left(\Bbb R^k/2\cdot\Bbb Z^k\right)\times\Bbb T^{\nu-k}$ inherited from
$\Bbb R^\nu$ and the operator $R_{\Cal A}$ multiplies the first $k$
coordinates by minus one, see also the previous section. According to Lemma
A.2.2 of [22], there are finitely many nonzero vectors
$w_1,\dots ,w_p\in\Bbb R^\nu$ (none of which is a scalar multiple of
another) --- not depending on the phase point $x_0$ --- such that
$\langle v_1^0-v_2^0;\, w_{j_1}\rangle=0$ for some $j_1$ (because of (i)),
and $\langle v_1^0-R_{\Cal A}v_2^0;\, w_{j_2}\rangle=0$ for some $j_2$
(because of (ii)). If at least one of the two projections
$P_{\Cal A}w_{j_1}$ and $P_{\Cal A}w_{j_2}$ is nonzero, then, as it follows
from the independence of the vectors $P_{\Cal A}(v_1^0-v_2^0)$ and
$P_{\Cal A}(v_1^0-R_{\Cal A}v_2^0)=P_{\Cal A}(v_1^0+v_2^0)$, the equations
$\langle v_1^0-v_2^0;\, w_{j_1}\rangle=0$ and
$\langle v_1^0-R_{\Cal A}v_2^0;\, w_{j_2}\rangle=0$ are independent, so they
together a define a manifold with codimension two. Such phase points $x_0$
are listed up in part (1) of the lemma.
Therefore, we can assume that $w_{j_1}$ and $w_{j_2}$ both belong to the
second factor $\Bbb R^{\nu-k}$ of the velocity space
$\Bbb R^\nu=\Bbb R^k\oplus\Bbb R^{\nu-k}$. If $w_{j_1}\ne w_{j_2}$
(i. e. they are not even parallel), then the equations
$\langle v_1^0-v_2^0;\, w_{j_1}\rangle=0$ and
$\langle v_1^0-R_{\Cal A}v_2^0;\, w_{j_2}\rangle=0$ are again independent,
and such phase points $x_0$ are listed up in (1).
The only way of obtaining a codimension-one fanily of exceptional phase
points $x_0$ is then to have $w_{j_1}=w_{j_2}$ and $P_{\Cal A}w_{j_1}=0$.
List up the vectors $w_j$ with $P_{\Cal A}w_{j}=0$ in a sequence
$z_1,\dots ,z_l$ ($\in\Bbb R^{\nu-k}$), and obtain the first statement of the
lemma.
The fact that for an exceptional phase point $x_0$ -- listed up in (1)-(2) --
the entire trajectory $S^{(-\infty,\infty)}x_0$ does not contain a single
ball-to-ball collision immediately follows from the following, simple
observation:
\medskip
\proclaim{Sublemma 4.2} Consider the two rays
$L^+,\, L^-\subset\Bbb R^n$ as follows:
$$
L^+=\left\{q_0+tv_0|\, t\ge 0\right\},
$$
$$
L^-=\left\{q_0+tv_0|\, t\le 0\right\},
$$
$q_0\in\Bbb R^n$, $0\ne v_0\in\Bbb R^n$. Then
$d(L^+,\, \Bbb Z^n)=d(L^-,\, \Bbb Z^n)$.
\endproclaim
\subheading{Proof} Apply the natural projection
$\pi:\, \Bbb R^n\rightarrow \Bbb R^n/\Bbb Z^n=\Bbb T^n$ to the half lines
$L^{\pm}$. It is well known that
$\text{Cl}\left(\pi L^+\right)=\text{Cl}\left(\pi L^-\right)=S$ is a coset
with respect to a subtorus of $\Bbb T^n$. (Here $\text{Cl}$ denotes the
closure.) This means that if $d(L^+,\, \Bbb Z^n)<\alpha$, then $S$ intersects
the open ball with radius $\alpha$ centered at zero in $\Bbb T^n$ and,
therefore, $d(L^-,\, \Bbb Z^n)<\alpha$, as well.
This finishes the proof of Sublemma 4.2 and, hence, the proof of Lemma 4.1,
as well. \qed
\medskip
\subheading{Remark} Part (2) of the assertion of 4.1 is really not vague.
Indeed, in the case of a trajectory without a ball-to-ball collision the time
evolutions of the $q$-components are {\it independent} of each other, thanks
to the orthogonality of the walls of the container. Therefore, if the
$\overline{\Cal A}$-component $P_{\overline{\Cal A}}(v_1^0-v_2^0)$ of the
initial relative velocity belongs to some exceptional hyperplane of
$\Bbb R^{\nu-k}$ (just as in the proof of Lemma A.2.2 of [22]) and
the the initial relative position $P_{\overline{\Cal A}}(q_1^0-q_2^0)$
belongs to some non-empty, open region, then even the
$\overline{\Cal A}$-parts of the positions $q_i^t$ ($i=1,2$) never get
closer to each other than $2r$, and there will be no ball-to-ball collision.
\medskip
Thus, by dropping the exceptional phase points $x_0$ listed up in (1) and (2)
of 4.1 (the latter set of phase points will be discussed in the next section),
we can assume that the non-singular trajectory $\traj$ contains infinitely
many ball-to-ball collisions in each time direction. Let us list all such
collisions of $\traj$ as $(\dots,\sigma_{-1},\sigma_0,\sigma_1,\dots)$, so
that $\sigma_0$ is the first collision occuring in positive time ($t=0$ is
supposed not to be a moment of collision), i. e. there is not even a
non-scattering, wall-to-ball collision in the time interval $[0,t_0)$.
Just as in the previous section, $t_n=t(\sigma_n)$ denotes the time of the
$n$-th collision, $n\in\Bbb Z$. Following (3.2), we can now speak about the
sets $Z_i(1)$, $Z_i(2)$, $Z_i\subset\Cal A$, ($i\in\Bbb Z$). The doubly
infinite sequence
$$
\Sigma=\left(\dots,\, \sigma_{-1},\, Z_0,\, \sigma_0,\, Z_1,\,
\sigma_1,\, \dots\right)
$$
is now called the symbolic collision sequence of the trajectory
$\omega=\traj$.
\medskip
{\bf From now on, in this section we will always be assuming the following
inductive hypothesis:}
\medskip
\proclaim{Hypothesis 4.3} For every integer $k'2r$
for every $t\in\Bbb R$, while
$d\left(\hat q_1^{t_n(x)}(x),\, R_{\{1\}}\hat q_2^{t_n(x)}(x)\right)=2r$
($n\in\Bbb Z$), where $t_n(x)=t\left(\sigma_n(x)\right)$ is the time of the
$n$-th scattering collision $\sigma_n$ on the orbit of $x\in U_0\cap P_1$.
The above discussed $\hat{\bold C}$-dynamics
$\left\{(\hat q_1^t(x),\, \hat q_2^t(x))|\, t\in\Bbb R\right\}$ (which is
a standard $(\nu,k-1,r)$-flow without the normalizations
$(v_1)_1+(v_2)_1=(q_1)_1+(q_2)_1=0$) can now be defined for every phase point
$x\in U_0$ by using the initial lifting
$(\hat q_1^0(x),\, \hat q_2^0(x))=(q_1^0(x),\, q_2^0(x))$ of the positions,
irrespectively of whether $x\in P_1$ or not. (In other words, when defining
this $\hat{\bold C}$-dynamics
$\left\{(\hat q_1^t(x),\, \hat q_2^t(x))|\, t\in\Bbb R\right\}$, we do not
remove the antipodal cylinder
$d\left(\hat q_1,\, R_{\{1\}}\hat q_2\right)<2r$ from the configuration
space but, rather, we allow the above inequalities.) We choose a small number
$\epsilon_0>0$ and define the following closed subsets of $U_0$ (see also
(5.2) and (5.7) of [11]):
$$
\aligned
F_+=\left\{x\in U_0\big|\, d\left(\hat q_1^t(x),\, R_{\{1\}}\hat q_2^t(x)
\right)\ge 2r \quad \forall \, t\ge 0\right\}, \\
F_-=\left\{x\in U_0\big|\, d\left(\hat q_1^t(x),\, R_{\{1\}}\hat q_2^t(x)
\right)\ge 2r \quad \forall \, t\le 0\right\},
\endaligned
\tag 4.7
$$
$$
\aligned
F'_+=\left\{x\in U_0\big|\, d\left(\hat q_1^t(x),\, R_{\{1\}}\hat q_2^t(x)
\right)\ge 2r-\epsilon_0 \quad \forall \, t\ge 0\right\}, \\
F'_-=\left\{x\in U_0\big|\, d\left(\hat q_1^t(x),\, R_{\{1\}}\hat q_2^t(x)
\right)\ge 2r-\epsilon_0 \quad \forall \, t\le 0\right\},
\endaligned
\tag 4.8
$$
where, in the case of singular trajectories, we understand these inequalities
in such a way that they should hold for {\it some} trajectory branch. This
convention makes the sets $F_{\pm}\subset U_0$, $F'_{\pm}\subset U_0$ closed,
just as in Main Lemma 5.1 of [11] or in (5.4)--(5.7) of
[12]. It is obvious that
$$
F_+\subset F'_+, \qquad F_-\subset F'_-,
\tag 4.9
$$
and
$$
U_0\cap P_1\subset F_-\cap F_+.
\tag 4.10
$$
Since the $\hat{\bold C}$-flow
$\left\{(\hat q_1^t(x),\, \hat q_2^t(x))|\, t\in\Bbb R\right\}$ has the
quantities $(v_1)_1+(v_2)_1$ and $(q_1)_1+(q_2)_1$ as first integrals, it is
now quite natural to define a foliation
$$
\aligned
U_0 &=\cup\left\{H_{a,b}|\quad a\in I_1,\; b\in I_2 \right\} \\
H_{a,b} &:=\left\{x\in U_0|\, (q_1(x))_1+(q_2(x))_1=a,\;
(v_1(x))_1+(v_2(x))_1=b \right\},
\endaligned
\tag 4.11
$$
where $I_1,\, I_2\subset\Bbb R$ are suitable open intervals. (We can assume
that the shape of the small, open neighborhood $U_0\ni x_0$ is such that it
permits us to establish the union (4.11) with open intervals $I_1$, $I_2$.)
It is an important consequence of the assumption 4.3/A that for $\mu$-almost
every phase point $x\in U_0$ there exist the maximum--dimensional
(actually, $(\nu+k-2)$-dimensional) local, exponentially stable and unstable
invariant manifolds $\gamma^s(x)$, $\gamma^u(x)$ (containing $x$ as an
interior point) with respect to the $\hat{\bold C}$-dynamics, defineable as
follows:
$$
\aligned
\gamma^s(x)=\text{CC}_x\left\{y\in U_0\big|\, d(\hat x^t,\hat y^t)\rightarrow
0 \text{ exp. fast as } t\rightarrow +\infty\right\}, \\
\gamma^u(x)=\text{CC}_x\left\{y\in U_0\big|\, d(\hat x^t,\hat y^t)\rightarrow
0 \text{ exp. fast as } t\rightarrow -\infty\right\},
\endaligned
\tag 4.12
$$
where
$$
\hat x^t=(\hat q_1^t(x),\hat q_2^t(x);\, \hat v_1^t(x),\hat v_2^t(x)),
$$
and the $\hat{\bold C}$-phase point $\hat y^t$ is defined analogously.
(Here the symbol $\text{CC}_x(.)$ denotes the connected component of a set
containing the point $x$.) Clearly,
$\gamma^s(x)\cup\gamma^u(x)\subset H_{a,b}$ for $x\in H_{a,b}$.
According to the fundamental ``integrability'' of slimness (see Section 2 or
Lemma 3.8 of [15]), it is enough to show that the closed subset
$F_-\cap F_+\cap H_{a,b}$ of $H_{a,b}$ is slim in $H_{a,b}$ for every
$a\in I_1$, $b\in I_2$. It follows immediately from the ergodicity assumption
of 4.3/A that
$$
\mu_{H_{a,b}}\left((F'_-\cup F'_+)\cap H_{a,b}\right)=0.
\tag 4.13
$$
Recall that $F'_{\pm}\supset F_{\pm}$. In order to show that the closed, zero
set $F_-\cap F_+\cap H_{a,b}$ has at least two codimensions in $H_{a,b}$,
it is enough to prove that this set enjoys the non-separating property, see
(ii) of Lemma 3.9 in [15]. This is what we will do.
Besides the exponentially stable and unstable manifolds
$\gamma^s(x),\, \gamma^u(x)\subset H_{a,b}$ ($x\in H_{a,b}$), we need to use
the one-dimensional neutral manifolds $\gamma^0(x)\subset H_{a,b}$
corresponding to the $\hat{\bold C}$-flow direction. We set
$$
\aligned
\gamma^0(x)=\big\{y\in H_{a,b}\big|\, v_i(y)=v_i(x)\text{ and }\exists
\lambda\in\Bbb R \text{ such that} \\
q_i(y)=q_i(x)+\lambda v_i(x)-
\frac{\lambda b}{2}(1,0,\dots ,0) \quad (i=1,2)\big\}
\endaligned
\tag 4.14
$$
for $x\in H_{a,b}$, $a\in I_1$, $b\in I_2$. Subtracting the vector
$\dfrac{\lambda b}{2}(1,0,\dots ,0)$ from the positions is just needed in
order to project back into $H_{a,b}$. It is clear that the following
transversality condition holds, see also (5.19) and 5.20 of [12]:
$$
\Cal T_x H_{a,b}=\Cal T_x\gamma^u(x)+\Cal T_x\gamma^s(x)+\Cal T_x\gamma^0(x),
\tag 4.15
$$
where $\Cal T_x(.)$ enotes the tangent space of a manifold at the foot point
$x$ and $+$ is a notation for the direct sum (not necessarily orthogonal)
of linear spaces.
It is easy to see that for every $x\in U_0$, $y\in\gamma^s(x)$ or
$y\in\gamma^0(x)$
$$
d\left(\hat q^t(x),\, \hat q^t(y)\right)\le d\left(\hat q^0(x),\, \hat
q^0(y)\right)\le\text{diam}U_0 \text{ for } t\ge 0,
\tag 4.16
$$
and, analogously, for any pair $x,y\in U_0$ for which $y\in\gamma^u(x)$ or
$y\in\gamma^0(x)$
$$
d\left(\hat q^t(x),\, \hat q^t(y)\right)\le d\left(\hat q^0(x),\, \hat
q^0(y)\right)\le\text{diam}U_0 \text{ for } t\le 0.
\tag 4.17
$$
Therefore, if the size $\text{diam}U_0$ is chosen small enough compared to
$\epsilon_0$, then we have the analogue of Lemma 5.8 from [11]:
\medskip
\proclaim{Sublemma 4.18} For any $x\in U_0$, if
$\gamma^s(x)\cap F_+\ne\emptyset$ ($\gamma^0(x)\cap F_+\ne\emptyset$),
then $\gamma^s(x)\subset F'_+$ ($\gamma^0(x)\subset F'_+$). Analogously, if
$\gamma^u(x)\cap F_-\ne\emptyset$ ($\gamma^0(x)\cap F_-\ne\emptyset$),
then $\gamma^u(x)\subset F'_-$ ($\gamma^0(x)\subset F'_-$).
\endproclaim
\medskip
According to our assumption 4.3, if the fixed base point $x_0$ does not
belong to some exceptional, slim subset of $\bold M$ determined by the
$\hat{\bold C}$-dynamics, then $x_0$ is sufficient with respect to this
$\hat{\bold C}$-flow and the fundamental statement of Lemma 3 of
[21] (or Theorem 3.6 of [10]) holds true. (Since their
formalism is too technical, we do not even quote them here.) As a direct
corollary of these results, the absolute continuity of the triple of
foliations $\gamma^s(.)$, $\gamma^u(.)$, $\gamma^0(.)$ (see Theorem 4.1
of [7]), and the transversality relation (4.15), we obtain the
crucial Zig-zag Lemma, the precise analogue of Corollary 3.10 of
[10]:
\medskip
\proclaim{Sublemma 4.19 (Zig-zag Lemma)} Suppose that the base point
$x_0\in\bold M$ is sufficient with respect to the $\hat{\bold C}$-dynamics.
Then we can select the open neighborhood $U_0\ni x_0$ so small that the
following assertion holds true:
\medskip
For every manifold $H_{a,b}\subset U_0$ ($a\in I_1$, $b\in I_2$), for every
open, connected subset $G\subset H_{a,b}$, and for almost every pair of points
$x,\, y\in G\setminus(F'_-\cup F'_+)$ one can find a finite sequence
$$
\xi_1^u,\; \xi_1^0,\; \xi_1^s,\; \xi_2^u,\; \xi_2^0,\; \xi_2^s,\dots
,\xi_n^u,\; \xi_n^0,\; \xi_n^s
$$
of submanifolds of $G$ with the following properties:
\medskip
(i) $\xi_i^\alpha$ is an open, connected subset of $\gamma^\alpha(z)$ for
some $z\in\xi_i^\alpha$, $\alpha=u,\, 0,\, s$, $i=1,\dots ,n$;
\medskip
(ii) $x\in\xi_1^u$, $y\in\xi_n^s$;
\medskip
(iii)
$$
\emptyset\ne\xi_i^u\cap\xi_i^0\subset G\setminus(F'_-\cup F'_+),\;
\emptyset\ne\xi_i^0\cap\xi_i^s\subset G\setminus(F'_-\cup F'_+),\;
\emptyset\ne\xi_i^s\cap\xi_{i+1}^u\subset G\setminus(F'_-\cup F'_+)
$$
for $i=1,\dots ,n$.
\endproclaim
\medskip
\subheading{Remark} The non-empty intersections in (iii) must be
one-point-sets, according to the transversality (4.15).
\medskip
It follows immediately from (i)--(iii) and Sublemma 4.18 that
$$
\bigcup\left\{\xi_i^\alpha \big|\, \alpha=u,\, 0,\, s;\quad i=1,\dots ,n
\right\}\subset G\setminus(F_-\cap F_+).
\tag 4.20
$$
Thus we obtained that almost every pair of points
$x,\, y\in G\setminus(F'_-\cup F'_+)$ can be connected by a continuous curve
$\left\{\xi(t)|\, 0\le t\le 1\right\}$ so that
$\xi(t)\in G\setminus(F_-\cap F_+)$ for $0\le t\le 1$. Since the open subset
$G\setminus(F'_-\cup F'_+)$ has full measure in $G$ (and, therefore, it is
dense in $G$), we get that the open subset $G\setminus(F_-\cap F_+)$ of
$G$ must be connected. This proves (ii) of Lemma 3.9 of [15],
i. e. $\text{dim}(F_-\cap F_+\cap H_{a,b})\le\text{dim}H_{a,b}-2$ and,
therefore, $F_-\cap F_+$ is indeed a slim subset of $U_0$.
The proof of Lemma 4.5 is now complete. \qed
\medskip
The last lemma of this section takes care of the case $k=\nu$,
$|Z_i|\left(\nu-|Z_i|\right)=0$, ($i\in\Bbb Z$).
\medskip
\proclaim{Lemma 4.21} Assume that $k=\nu$ and
$|Z_i|\left(\nu-|Z_i|\right)=0$ for every integer $i$. Then the phase point
$x_0$ necessarily belongs to some slim, exceptional subset of $\bold M$.
\endproclaim
\medskip
\subheading{Proof (A sketch)} Since the proof of this lemma is very
similar to that of Lemma 4.5, we are not going to present it here in whole
detail but, instead, we will just point out the differences between the two
proofs.
We again begin with ``unfolding'' the trajectory
$$
\traj=\left\{\left(q_1^t(x_0),\, q_2^t(x_0)\right)\big|\; t\in\Bbb R\right\}
\tag 4.22
$$
by reflecting the cubic container $\bold C=[0,1]^\nu$ across its faces
$(q)_j=0$ and $(q)_j=1$, $j=1,\dots ,\nu$, pretty much the same way as we did
in the proof of 4.5. Namely, we consider the covering map
$$
\Phi=(\phi,\dots ,\phi):\; \hat{\bold C}=\Bbb R^\nu/2\cdot\Bbb Z^\nu
\rightarrow [0,1]^\nu=\bold C
\tag 4.23
$$
with the rooftop map $\phi:\, \Bbb R/2\cdot\Bbb Z\rightarrow[0,1]$,
$\phi(y):=d(y,\, 2\cdot\Bbb Z)$ acting on all components, see also (4.6).
Then we just pull back the trajectory of (4.22) by the mapping $\Phi$
to obtain the unfolded trajectory
$$
\hat\omega(x_0)=\left\{\left(\hat q_1^t(x_0),\, \hat q_2^t(x_0)\right)
\big|\; t\in\Bbb R\right\},
$$
$\hat q_i^t(x_0)\in\Bbb R^\nu/2\cdot\Bbb Z^\nu$ by using the selected
initial pull-back
$$
\left(\hat q_1^0(x_0),\, \hat q_2^0(x_0)\right)=
\left(q_1^0(x_0),\, q_2^0(x_0)\right), \quad \text{mod }2\cdot\Bbb Z^\nu.
$$
The hypothesis $|Z_i|\left(\nu-|Z_i|\right)=0$ ($i\in\Bbb Z$) implies that the
obtained unfolded curve
$$
\hat\omega\subset\left(\Bbb R^\nu/2\cdot\Bbb Z^\nu\right)\times
\left(\Bbb R^\nu/2\cdot\Bbb Z^\nu\right)=\hat{\bold C}\times\hat{\bold C}
$$
has reflections only at the cylinder
$$
C=\left\{(\hat q_1,\, \hat q_2)|\; d(\hat q_1,\, \hat q_2)=2r\right\}
$$
and the ``antipodal'' cylinder
$$
\overline{C}=
\left\{(\hat q_1,\, \hat q_2)|\; d(\hat q_1,\, -\hat q_2)=2r\right\}.
$$
We can now define the unfolded orbit
$$
\hat\omega(x)=\left\{\left(\hat q_1^t(x),\, \hat q_2^t(x)\right)
\big|\; t\in\Bbb R\right\}
$$
for every phase point $x=\left(q_1^0(x),\, q_2^0(x)\right)=
\left(\hat q_1^0(x),\, \hat q_2^0(x)\right)\in U_0$ by forgetting about the
collisions at $d\left(\hat q_1,\, R_Z\hat q_2\right)=2r$ with
$0<|Z|<\nu$ (making these cylinders transparent), and retaining only the
collisions at the cylinders $C$ and $\overline{C}$. In the Appendix we show
that the so obtained $\hat{\bold C}$-dynamics
($\hat{\bold C}=\Bbb R^\nu/2\cdot\Bbb Z^\nu$) with the configuration space
$\hat{\bold C}\times\hat{\bold C}\setminus(C\cup\overline{C})$ is, in fact,
a splitting orthogonal cylindric billiard in the sense of the article
[22]. There is now one additional first integral, namely the energy
$\Vert \hat v_1+\hat v_2\Vert^2$. Therefore, the foliation of the small open
neighborhood $U_0\ni x_0$ (the analogue of (4.11)) is now going on by
specifying the value of
$\Vert \hat v_1^0+\hat v_2^0\Vert^2=\Vert v_1^0+v_2^0\Vert^2$:
$$
\aligned
U_0 &=\bigcup\Sb a\in I\endSb H_a, \\
H_a &:=\left\{x\in U_0 \big|\; \Vert v_1^0(x)+v_2^0(x)\Vert^2=a \right\}.
\endaligned
\tag 4.24
$$
It is shown in the Appendix that, for any fixed value of
$\Vert\hat v_1+\hat v_2\Vert^2$, the $\hat{\bold C}$-flow is the Cartesian
product of two dispersive Sinai--billiards and, therefore, it is Bernoulli,
and all but two of its Lyapunov exponents are nonzero. Thus, in this situation
we have $\text{dim}\gamma^s(x)=\text{dim}\gamma^u(x)=2\nu-2$,
$\text{dim}\gamma^0(x)=2$, $\text{dim}H_a=4\nu-2$, and for every $x\in U_0$
the linear direct sum of the tangent spaces of the transversal
$\gamma^s(x)$, $\gamma^u(x)$ and $\gamma^0(x)$ is exactly the tangent space
of the folium $H_a\ni x$ through $x$, see also (4.15). If the original orbit
$\omega(x)=S^{(-\infty,\infty)}x$ of a phase point $x\in U_0$ has the property
$|Z_i(x)|\left(\nu-|Z_i(x)|\right)=0$ ($i\in\Bbb Z$), then this fact is
reflected by the $\hat{\bold C}$-orbit $\hat\omega(x)$ in such a way that the
configuration point $\left(\hat q_1^t(x),\, \hat q_2^t(x)\right)$ never enters
a ``forbidden'' open region, namely the interiors of the cylinders
$d\left(\hat q_1,\, R_Z\hat q_2\right)\le 2r$, $0<|Z|<\nu$. This makes it
possible to define the closed sets $F_+$, $F_-$ just the same way as in the
proof of 4.5. Also, by shrinking the forbidden region mentioned above, we can
define the larger closed sets $F'_+$, $F'_-$. Thanks to the ergodicity of
the $\hat{\bold C}$-flow, we again have
$$
\mu_{H_a}\left((F'_-\cup F'_+)\cap H_a\right)=0.
$$
Plainly,
the analogues of sublemmas 4.18--4.19 remain valid in the recent situation.
By putting together these ingredients, we see that the closed set
$F_-\cap F_+\cap H_a$ has, indeed, at least two codimensions in $H_a$.
Finally, by using again the integrability of slimness (Lemma 3.8 of
[15]), we obtain that the points $x\in U_0$ with the property
$|Z_i(x)|\left(\nu-|Z_i(x)|\right)=0$ ($i\in\Bbb Z$) form a slim subset
of $U_0$. This finishes the proof of Lemma 4.21 and, therefore, the proof of
Key Lemma 4.4, as well. \qed
\bigskip \bigskip
\centerline{5. Proof of the Theorem}
\bigskip \bigskip
By using the results of the preceding sections, here we prove our
\medskip
\proclaim{Theorem} The standard $(\nu,k,r)$-flow
($\nu\ge 2$, $0\le k\le\nu$, $0n \right\}.
\endaligned
\tag 5.2
$$
(If the positive orbit $S^{[0,\infty)}x$ is singular, then we understand the
above requirements in such a way that they should hold for {\bf some} branch
of $S^{[0,\infty)}x$, thus making the sets $F_+^{(1)}$, $F_+^{(2)}$ closed
in $\hat J_1$.)
We claim that both sets $F_+^{(1)}$, $F_+^{(2)}$ have an empty interior in
$\hat J_1$.
\endproclaim
\medskip
\subheading{Remark 5.3} In the case $k<\nu$ the set $F_+^{(2)}$ is
contained in $F_+^{(1)}$ (meaning $\cup_{l>n} Z_l(x)=\emptyset$ for every
$x\in F_+^{(2)}$), and the statement about $F_+^{(2)}$ follows from the claim
about $F_+^{(1)}$. The result $\text{int}_{\hat J_1} F_+^{(2)}=\emptyset$
is only significant in the case $k=\nu$.
\medskip
\subheading{Proof} Since the proof is very similar to the proof of
Lemma 6.1 of [15], here we will only present a rough sketch of it.
Furthermore, the proof of $\text{int}_{\hat J_1} F_+^{(2)}=\emptyset$
(in the case $k=\nu$) is quite analogous to the proof of
$\text{int}_{\hat J_1} F_+^{(1)}=\emptyset$ and, therefore, we will only be
dealing with the result
$\text{int}_{\hat J_1} F_+^{(1)}=\emptyset$. (This analogy is similar to the
analogy between the proofs of lemmas 4.5 and 4.21.)
Without restricting the generality, we assume that $j_2=1$. We argue by
contradiction, so we suppose that
$\text{int}_{\hat J_1} F_+^{(1)}\ne\emptyset$ and, by choosing a smaller open
piece of the manifold $\hat J_1$, we can assume that $F_+^{(1)}=\hat J_1$,
i. e. the positive orbit $S^{[0,\infty)}x$ of every point $x\in\hat J_1$
follows a pretty irregular pattern: $1\notin\cup_{l=n+1}^\infty Z_l(x)$.
Select now a base point $x_0\in\hat J_1$ and a small, open ball neighborhood
$U_0\subset\bold M(\nu,k,r)$ of $x_0$. We can assume that the time point
$b>0$ is chosen in such a way that there is no collision (not even a wall
collision) between $t_n(x)=t\left(\sigma_n(x)\right)$ and $b$ for every
$x\in U_0$.
Let us push forward the open set $U_0$ and $U_0\cap\hat J_1$ by the $b$-th
power of the $(\nu,k,r)$-flow: $S^b(U_0)=W_0$,
$S^b\left(U_0\cap\hat J_1\right)=K_0$, $S^b(x_0)=y_0$. By our assumption,
all trajectories $S^{(0,\infty)}y$ follow the irregular pattern
$$
1\notin\bigcup_{l=0}^\infty Z_l(y), \quad y\in K_0.
\tag 5.4
$$
Therefore, for every $y\in K_0$ we can again`` unfold'' the positive orbit
$$
S^{(0,\infty)}y=\left\{\left(q_1^t(y),\, q_2^t(y)\right)|\, t\ge 0\right\}
$$
by reflecting our container $\bold C=[0,1]^k\times\Bbb T^{\nu-k}$ across two
of its faces $(q)_1=0$, $(q)_1=1$, and obtain the positive orbit
$$
\hat{S}^{(0,\infty)}y=\left\{\left(\hat q_1^t(y),\,
\hat q_2^t(y)\right)|\; t\ge 0\right\}
$$
in the container
$$
\hat{\bold C}=\left(\Bbb R/2\cdot\Bbb Z\right)\times [0,1]^{k-1}\times
\Bbb T^{\nu-k},
$$
$\hat q_i^t(y)\in\hat{\bold C}$, see also the proof of Lemma 4.5.
Quite similarly to the proof of Lemma 4.5, the above defined
$\hat{\bold C}$-dynamics
$$
\hat\omega(y)=\left\{\left(\hat q_1^t(y),\,
\hat q_2^t(y)\right)|\; t\ge 0\right\} \quad (y\in K_0),
$$
which is essentially a standard $(\nu,k-1,r)$-flow without the normalization
$(v_1)_1+(v_2)_1=(q_1)_1+(q_2)_1=0$, can now be defined for every phase point
$y\in W_0$ by using the initial lifting
$\left(\hat q_1^0(y),\, \hat q_2^0(y)\right)=\left(q_1^0(y),\,q_2^0(y)\right)$
of the positions, irrespectively of whether
$1\in\cup_{l=0}^\infty Z_l(y)$, or not.
Consider now the codimension-two submanifold $\tilde W_0\subset W_0$
$$
\aligned
\tilde W_0= &\big\{y\in W_0\big|\;\left(q_1(y)\right)_1+\left(q_2(y)\right)_1=
\left(q_1(y_0)\right)_1+\left(q_2(y_0)\right)_1 \\
&\text{and } \left(v_1(y)\right)_1+\left(v_2(y)\right)_1=
\left(v_1(y_0)\right)_1+\left(v_2(y_0)\right)_1 \big\},
\endaligned
\tag 5.5
$$
and the intersection $\tilde K_0=K_0\cap\tilde W_0$, corresponding to fixing
the values of $(q_1)_1+(q_2)_1$ and $(v_1)_1+(v_2)_1$ according to what these
values are for the base point $y_0=S^bx_0\in K_0$. The phase point
$y\in\tilde W_0$ and its positive $\hat{\bold C}$-orbit $\hat\omega(y)$
can be naturally identified with the phase point and positive orbit of a
standard $(\nu,k-1,r)$-flow by changing the reference coordinate system, i. e.
by reducing the first component of $q_1+q_2$ and $v_1+v_2$ to zero. Although
we will be using this equivalence, but, for the sake of brevity, we are not
going to introduce an extra notation for that purpose.
The meaning of (5.4) is that there exists a small, open ball
$B(z_0,\epsilon_0)\subset\bold M(\nu,k-1,r)$ of the phase space of the
$(\nu,k-1,r)$-flow, such that for every $y\in\tilde K_0$ the positive
$\hat{\bold C}$-orbit
$$
\hat\omega(y)=\left\{\hat y^t=\left(\hat q_1^t(y),\,
\hat q_2^t(y)\right)|\; t\ge 0\right\}
$$
avoids the ball $B(z_0,\epsilon_0)$.
We can now define the exponentially stable, local invariant manifolds
$\gamma^s(x)\subset\tilde W_0$ for almost every phase point $x\in\tilde W_0$,
similarly to (4.12), as follows:
$$
\gamma^s(x)=\text{CC}_x\left\{y\in\tilde W_0\big|\, d(\hat x^t,\hat y^t)
\rightarrow 0\text{ exp. fast as }t\rightarrow +\infty\right\}.
$$
\medskip
\proclaim{Sublemma 5.6} The codimension-one submanifold $\tilde K_0$ of
$\tilde W_0$ is transversal to the invariant manifold
$\gamma^s(y)\subset\tilde W_0$ for every $y\in\tilde K_0$, whenever
$\gamma^s(y)$ is a manifold containing $y$ as an interior point.
\endproclaim
\medskip
\subheading{Proof} This transversality immediately follows from our
combinatorial assumption ($j_1\notin Z_0(x)$ or $j_1\in\cup_{l=1}^n Z_l(x)$
for every $x\in U_0$) by the method of the proof of Lemma 3.10. We note that
this is just the point of the proof of 5.1 where we use the above mentioned
combinatorial hypothesis. \qed
It follows now from the transversality sublemma and from the Transversal
Fundamental Theorem for semi-dispersive billiards (Theorem 3.6 of
[10]) that for almost every phase point $y\in\tilde K_0$
(with respect to any smooth measure on $y\in\tilde K_0$) the set
$\gamma^s(y)$ is a submanifold of $\tilde W_0$ containing $y$ as an
interior point. Therefore, the union
$$
E:=\bigcup\Sb y\in\tilde K_0\endSb \gamma^s(y)\subset\tilde W_0
$$
has positive measure in $\tilde W_0$. However, if the size of the open set
$W_0$ is chosen small enough compared to the radius $\epsilon_0$ of the
avoided ball $B(z_0,\epsilon_0)$, then the already proved ball avoiding
$\hat\omega(y)\cap B(z_0,\epsilon_0)=\emptyset$ implies that the positive
$\hat{\bold C}$-orbit $\hat\omega(y')$ of any $y'\in\gamma^s(y)$ avoids the
shrunk ball $B(z_0,\epsilon_0/2)$, see also (4.16) and Sublemma 4.18.
Thus, we obtained that in the $(\nu,k-1,r)$-dynamics the positive trajectory
of every phase point $y\in E$ avoids the open ball $B(z_0,\epsilon_0/2)$,
and, yet $E$ has positive measure in the phase space $\bold M(\nu,k-1,r)$.
This contradicts our inductive hypothesis 4.3 postulating the ergodicity of
the $(\nu,k-1,r)$-flow. Hence Lemma 5.1 follows. \qed
\medskip
\proclaim{Corollary 5.7} Assume the inductive Hypothesis 4.3. Then there
exists a slim subset $\hat S\subset\bold M=\bold M(\nu,k,r)$ of the phase
space of the standard $(\nu,k,r)$-flow such that every phase point
$x\in\bold M\setminus\hat S$ either belongs to the union $\Cal F$ of the
codimension-one submanifolds featuring in part (2) of Lemma 4.1, or the
trajectory $S^{(-\infty,\infty)}x$ contains at most one singularity and
it is {\bf sufficient}. Furthermore, the Chernov--Sinai Ansatz
(see Section 2) holds true for the standard $(\nu,k,r)$-flow.
\endproclaim
\medskip
\subheading{Proof} Lemma 4.1 says that for every phase point
$x\in\bold M\setminus(\hat S_1\cup\Cal F)$ ($\hat S_1,\hat S_2,\dots$
will always denote some slim subsets of $\bold M$) the orbit
$S^{(-\infty,\infty)}x$ contains infinitely many scattering (ball-to-ball)
collisions in each time direction. Furthermore, thanks to Lemma 4.1 of
[10], for every phase point
$x\in\bold M\setminus(\hat S_2\cup\Cal F)$ ($\hat S_2\supset\hat S_1$)
the trajectory $S^{(-\infty,\infty)}x$ contains at most one singularity.
Then key lemmas 4.4, 3.5 and lemmas 5.1, 3.10 imply that every non-singular
phase point
$x\in\bold M\setminus(\hat S_3\cup\Cal F)$ ($\hat S_3\supset\hat S_2$)
has a sufficient orbit $S^{(-\infty,\infty)}x$.
As far as the singular phase points $x\in\Cal S\Cal R^+$ are concerned,
Lemma 6.1 of [15] asserts that for a generic (with respect to the
surface measure of $\Cal S\Cal R^+$) singular phase point
$x\in\Cal S\Cal R^+$ the positive orbit $S^{(0,\infty)}x$ is non-singular
and sufficient. This proves that
\medskip
(i) for every phase point
$x\in\bold M\setminus(\hat S_4\cup\Cal F)$ ($\hat S_4\supset\hat S_3$)
the orbit $S^{(-\infty,\infty)}x$ contains at most one singularity, and
it is sufficient;
\medskip
(ii) the Chernov--Sinai Ansatz holds true.
This proves Corollary 5.7. \qed
\medskip
\proclaim{Corollary 5.8} Assume Hypothesis 4.3. Denote by
$\bold M\setminus\Cal F=\bigcup_{i=1}^r \Omega_i$ the decomposition of the
open set $\bold M\setminus\Cal F$ into its connected components, where
$$
\Cal F=\left\{(q_1,q_2;\, v_1,v_2)\in\bold M \big|\,
P_{\overline{\Cal A}}(v_1-v_2)\perp z_j \text{ for some } j\le l \right\},
$$
see part (2) of Lemma 4.1.
We claim that every component $\Omega_i$ ($i=1,\dots ,r$) belongs to one
ergodic component of the standard $(\nu,k,r)$-flow.
\endproclaim
\medskip
\subheading{Remark} We note that in the case $k=\nu$ the set $\Cal F$
is empty ($l=0$) and, therefore, $r=1$.
\medskip
\subheading{Proof} Since the complement set $\Omega_i\setminus\hat S_4$
is connected and it has full measure in $\Omega_i$, the statement immediately
follows from the previous corollary and the Chernov--Sinai Theorem on Local
Ergodicity, i. e. Theorem 5 of [21]. \qed
\medskip
The last outstanding task in the inductive proof of our Theorem is to connect
the open components $\Omega_1,\dots ,\Omega_r$ by (positive beams of)
trajectories. This will be done by the method from the closing part of the
proof of Lemma 4 of [9]. Namely, it is enough to construct for
every vector $z_j\in\Bbb R^{\nu-k}$ a piece of trajectory connecting a phase
point $x^{(1)}=\left(q_1^{(1)},q_2^{(1)};\, v_1^{(1)},v_2^{(1)}\right)$
with the property $\langle v_1^{(1)}-v_2^{(1)};\, z_j\rangle>0$ by another
phase point $x^{(2)}=\left(q_1^{(2)},q_2^{(2)};\, v_1^{(2)},v_2^{(2)}\right)$
for which $\langle v_1^{(2)}-v_2^{(2)};\, z_j\rangle<0$. This can be done,
however, very easily by slightly perturbing a tangential $(1,2)$-collision
for which the normal vector of impact is parallel with the vector $z_j$
and, therefore, the relative velocity of the tangentially ``colliding''
balls is perpendicular to $z_j$. Plainly, we can perturb this tangential
collision in such a way that the perturbed collision is no longer singular,
and the normal vector of impact does not change. In this way we obtain the
required phase points $x^{(1)}$ and $x^{(2)}$ together with a piece of
trajectory connecting them.
This completes the inductive proof of our theorem. \qed
\bigskip \bigskip
\centerline{APPENDIX}
\bigskip
\centerline{A Special Orthogonal Cylindric Billiard}
\bigskip \bigskip
In the proof of Lemma 4.21 we unfolded the non-singular orbit of (4.22)
(with $|Z_i|(\nu-|Z_i|)=0$ for $i=0,\pm 1,\pm 2,\dots$) and thus obtained
the other trajectory
$$
\hat\omega(x_0)=\left\{\left(\hat q_1^t(x_0),\,
\hat q_2^t(x_0)\right)|\; t\in\Bbb R\right\},
$$
with $\hat q_i^t(x_0)\in\Bbb R^\nu/2\cdot\Bbb Z^\nu=\Bbb T_2^\nu$,
$i=1,2$. The so constructed orbit $\hat\omega(x_0)$ has collisions
only at the cylinder
$$
C=\left\{\left(\hat q_1,\, \hat q_2\right)\in\Bbb T_2^\nu \times
\Bbb T_2^\nu\big|\; d\left(\hat q_1,\, \hat q_2\right)=2r \right\}
$$
and at the ``antipodal'' cylinder
$$
\overline{C}=
\left\{\left(\hat q_1,\, \hat q_2\right)\in\Bbb T_2^\nu \times
\Bbb T_2^\nu\big|\; d\left(\hat q_1,\, -\hat q_2\right)=2r \right\}.
$$
We need to understand the ergodic properties of the arising billiard
flow in the configuration space
$\bold Q=\Bbb T_2^\nu \times\Bbb T_2^\nu\setminus(C\cup\overline{C})$.
For the sake of simplifying the notations, we will work with the unit
torus $\Bbb T^\nu=\Bbb R^\nu/\Bbb Z^\nu$, instead of $\Bbb T_2^\nu$,
and we will supress $\hat{\;}$ from over $q$ and $v=\dot q$.
It turns out that the $\nu$-dimensional generator (constituent) spaces
$$
\Bbb T_+=\left\{(x,\, x)|\; x\in\Bbb T^\nu\right\}, \quad
\Bbb T_-=\left\{(y,\, -y)|\; y\in\Bbb T^\nu\right\}
$$
of the cylinders $C$ and $\overline{C}$ are obviously orthogonal.
Thus, our system is an {\it orthogonal cylindric billiard} in the sense
of [22]. Let us write the configuration point
$$
(q_1,\, q_2)\in\Bbb T^\nu\times\Bbb T^\nu\setminus\left(C\cup
\overline{C}\right)
$$
in the form $(q_1,\, q_2)=(x,\, x)+(y,\, -y)$. Since the group homomorphism
$\Psi:\, \Bbb T_+\times\Bbb T_-\rightarrow \Bbb T^\nu\times\Bbb T^\nu$,
$\Psi\left((x,x),\, (y,-y)\right)=(x+y,\, x-y)$ is surjective with the
kernel $\text{Ker}\Psi=\left\{\left((x,x),\, (x,x)\right)|\, 2x=0\right\}$,
we obtain a $2^\nu$-to-$1$ covering of the original flow if we switch from
$(q_1,\, q_2)\in\Bbb T^\nu\times\Bbb T^\nu$ to
$\left((x,x),\, (y,-y)\right)$ with $q_1=x+y$, $q_2=x-y$. In terms of
$x$ and $y$ the conditions $d(q_1,q_2)\ge 2r$ and $d(q_1,-q_2)\ge 2r$
precisely mean that $d(y,G)\ge r$ and $d(x,G)\ge r$, where
$G:=\left\{g\in\Bbb T^\nu|\, 2g=0\right\}$. There are now two first integrals:
$E_1=\dfrac{1}{2}\Vert\dot x\Vert^2=\dfrac{1}{8}\Vert\dot q_1+\dot q_2\Vert^2$
and $E_2=\dfrac{1}{2}\Vert\dot y\Vert^2=\dfrac{1}{8}\Vert\dot q_1-
\dot q_2\Vert^2$. Thanks to the orthogonality, the $(x,\, \dot x)$
and $(y,\, \dot y)$ parts of the covering dynamics evolve independently of
each other. This means that --- after fixing the values of
$\Vert\dot q_1+\dot q_2\Vert^2$ and $\Vert\dot q_1-\dot q_2\Vert^2$ ---
the $2^\nu$-to-$1$ covering flow is the product of two, $\nu$-dimensional,
dispersive Sinai billiards. Therefore, the original orthogonal billiard
flow is mixing and all but two of its Lyapunov exponents are nonzero.
Besides that, for almost every phase point
$z=(q_1,q_2;\, \dot q_1,\dot q_2)$ of this orthogonal cylindric billiard
(with fixed values of
$\Vert\dot q_1+\dot q_2\Vert^2$ and $\Vert\dot q_1-\dot q_2\Vert^2$, of
course) the $(2\nu-2)$-dimensional, exponentially contracting and
expanding manifolds $\gamma^s(z)$ and $\gamma^u(z)$ exist, and they contain
$z$ as an interior point.
\bigskip \bigskip
\subheading{Acknowledgement} A significant part of this work was done
while the author enjoyed the warm hospitality and inspiring research
atmosphere at the Department of Mathematics of The Pennsylvania State
University, University Park Campus.
\bigskip \bigskip
\Refs
\bigskip \bigskip
\widestnumber\key{23}
\ref\key 1
\by L. Bunimovich, C. Liverani, A. Pellegrinotti, Yu. Sukhov
\paper Special Systems of Hard Balls that Are Ergodic
\jour Commun. Math. Phys.
\vol 146
\pages 357-396 (1992)
\endref
\medskip
\ref\key 2
\by L. A. Bunimovich, Ya. G. Sinai
\paper The fundamental theorem of the theory of scattering billiards
\jour Math. USSR-Sb. \vol 19 \pages 407-423 (1973)
\endref
\medskip
\ref\key 3
\by N. I. Chernov, C. Haskell
\paper Nonuniformly hyperbolic K-systems are Bernoulli
\jour Ergodic Theory and Dynamical Systems
\vol 16
\pages 19-44 (1996)
\endref
\medskip
\ref\key 4
\by R. Engelking
\paper Dimension Theory
\jour North Holland, 1978
\endref
\medskip
\ref\key 5
\by G. Gallavotti, D. Ornstein
\paper Billiards and Bernoulli schemes
\jour Commun. Math. Phys.
\vol 38
\pages 83-101 (1974)
\endref
\medskip
\ref\key 6
\by G. Galperin
\paper On systems of locally interacting and repelling particles moving in
space
\jour Trudy MMO
\vol 43
\pages 142-196 (1981)
\endref
\medskip
\ref\key 7
\by A. Katok, J.-M. Strelcyn
\paper Invariant Manifolds, Entropy and Billiards; Smooth Maps with
Singularities
\jour Lecture Notes in Mathematics
\vol 1222
\pages Springer, 1986
\endref
\medskip
\ref\key 8
\by I. P. Kornfeld, Ya. G. Sinai, S. V. Fomin
\paper Ergodic Theory
\jour Nauka, Moscow, 1980
\endref
\medskip
\ref\key 9
\by A. Kr\'amli, N. Sim\'anyi, D. Sz\'asz
\paper Ergodic Properties of Semi--Dispersing Billiards I.
Two Cylindric Scatterers in the 3--D Torus
\jour Nonlinearity
\vol 2
\pages 311--326 (1989)
\endref
\medskip
\ref\key 10
\by A. Kr\'amli, N. Sim\'anyi, D. Sz\'asz
\paper A ``Transversal'' Fundamental Theorem for Semi-Dis\-pers\-ing Billiards
\jour Commun. Math. Phys.
\vol 129
\pages 535--560 (1990)
\endref
\medskip
\ref\key 11
\by A. Kr\'amli, N. Sim\'anyi, D. Sz\'asz
\paper The K--Property of Three Billiard Balls
\jour Annals of Mathematics
\vol 133
\pages 37--72 (1991)
\endref
\medskip
\ref\key 12
\by A. Kr\'amli, N. Sim\'anyi, D. Sz\'asz
\paper The K--Property of Four Billiard Balls
\jour Commun. Math. Phys.
\vol 144
\pages 107-148 (1992)
\endref
\medskip
\ref\key 13
\by N. S. Krylov
\paper The Processes of Relaxation of Statistical Systems and the
Criterion of Mechanical Instability
\jour Thesis
\pages Moscow, (1942)
\moreref
\paper Republished in English by Princeton University Press
\pages Princeton N. J. (1979)
\endref
\medskip
\ref\key 14
\by D. Ornstein, B. Weiss
\paper On the Bernoulli Nature of Systems with Some Hyperbolic Structure
\jour Ergodic Theory and Dynamical Systems (accepted for publication)
\pages pp. 23 (1996)
\endref
\medskip
\ref\key 15
\by N. Sim\'anyi
\paper The K-property of $N$ billiard balls I
\jour Invent. Math.
\vol 108
\pages 521-548 (1992)
\endref
\medskip
\ref\key 16
\by N. Sim\'anyi
\paper The K-property of $N$ billiard balls II
\jour Invent. Math.
\vol 110
\pages 151-172 (1992)
\endref
\medskip
\ref\key 17
\by N. Sim\'anyi, D. Sz\'asz
\paper Hard Ball Systems Are Fully Hyperbolic
\jour Manuscript
\pages 1996
\endref
\medskip
\ref\key 18
\by Ya. G. Sinai
\paper On the Foundation of the Ergodic Hypothesis for a Dynamical
System of Statistical Mechanics
\jour Soviet Math. Dokl.
\vol 4
\pages 1818-1822 (1963)
\endref
\medskip
\ref\key 19
\by Ya. G. Sinai
\paper Dynamical Systems with Elastic Reflections
\jour Russian Math. Surveys
\vol 25:2
\pages 137-189 (1970)
\endref
\medskip
\ref\key 20
\by Ya. G. Sinai
\paper Development of Krylov's Ideas
\jour Afterword to N. S. Krylov's ``Works on the foundations of statistical
physics'', see reference [13]
\pages Princeton University Press (1979)
\endref
\medskip
\ref\key 21
\by Ya. G. Sinai, N.I. Chernov
\paper Ergodic properties of certain systems of 2--D discs and 3--D balls
\jour Russian Math. Surveys
\vol (3) 42
\pages 181-207 (1987)
\endref
\medskip
\ref\key 22
\by D. Sz\'asz
\paper The K-property of `Orthogonal' Cylindric Billiards
\jour Commun. Math. Phys.
\vol 160
\pages 581-597 (1994)
\endref
\medskip
\ref\key 23
\by L. N. Vaserstein
\paper On Systems of Particles with Finite Range and/or Repulsive
Interactions
\jour Commun. Math. Phys.
\vol 69
\pages 31-56 (1979)
\endref
\endRefs
\enddocument
**