Content-Type: multipart/mixed; boundary="-------------0210171901878"
This is a multi-part message in MIME format.
---------------0210171901878
Content-Type: text/plain; name="02-428.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="02-428.keywords"
Ergodicity, hyperbolicity, mathematical billiards, invariant foliation, hard ball systems, mixing
---------------0210171901878
Content-Type: application/x-tex; name="bolsin.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="bolsin.tex"
\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\endrem{}
\define\harmas{\left(\Sigma,\Cal A,\vec\tau\right)}
\define\harmasv{\left(\Sigma',\Cal A',\vec\tau'\right)}
\define\neutr{\Cal N(\omega)}
\define\dimc{\text{dim}_{\Bbb C}}
\noindent
October 17, 2002
\bigskip \bigskip
\heading
Proof of the Ergodic Hypothesis \\
for Typical Hard Ball Systems
\endheading
\bigskip \bigskip
\centerline{{\bf N\'andor Sim\'anyi}
\footnote{Research supported by the National Science Foundation, grant
DMS-0098773.}}
\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
\hbox{\centerline{\vbox{\hsize 8cm {\bf Abstract.} We consider the system of
$N$ ($\ge2$) hard balls with masses $m_1,\dots,m_N$ and radius $r$ in the
flat torus $\Bbb T_L^\nu=\Bbb R^\nu/L\cdot\Bbb Z^\nu$ of size $L$, $\nu\ge3$.
We prove the ergodicity (actually, the Bernoulli mixing property) of such
systems for almost every selection $(m_1,\dots,m_N;\,L)$ of the outer
geometric parameters. This theorem complements my earlier result that proved
the same, almost sure ergodicity for the case $\nu=2$. The method of that
proof was primarily dynamical-geometric, whereas the present approach is
inherently algebraic.}}}
\bigskip \bigskip
\noindent
Primary subject classification: 37D50
\medskip
\noindent
Secondary subject classification: 34D05
\bigskip \bigskip
\heading
\S1. Introduction
\endheading
\bigskip \bigskip
Hard ball systems or, a bit more generally, mathematical billiards
constitute an important and quite interesting family of dynamical systems
being intensively studied by dynamicists and researchers of mathematical
physics, as well. These dynamical systems pose many challenging mathematical
questions, most of them concerning the ergodic (mixing) properties of such
systems. The introduction of hard ball systems and the first major steps in
their investigations date back to the 40's and 60's, see Krylov's paper
[K(1942)] and Sinai's ground-breaking works [Sin(1963)] and [Sin(1970)],
in which the author --- among other things --- formulated the modern version
of Boltzmann's ergodic hypothesis (what we call today the Boltzmann--Sinai
ergodic hypothesis) by claiming that every hard ball system in a flat torus
is ergodic, of course after fixing the values of the trivial flow-invariant
quantities. In the articles
[Sin(1970)] and [B-S(1973)] Bunimovich and Sinai proved
this hypothesis for two hard disks on the two-dimensional unit torus
$\Bbb T^2$. The generalization of this result to higher dimensions $\nu>2$
took fourteen years, and was done by Chernov and Sinai in [S-Ch(1987)].
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 concave from
outside $\bold Q$. (They are bending away from $\bold Q$.)
The billiard systems of more than two hard balls in
$\Bbb T^\nu$ are no longer strictly dispersive, but just semi-dispersive
(strict concavity of the smooth components of $\partial\bold Q$
is lost, merely concavity 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 [K-S-Sz(1989)], [K-S-Sz(1990)],
[K-S-Sz(1991)], and [K-S-Sz(1992)]
we developed several new methods, and proved the ergodicity of
more and more complicated semi-dispersive billiards culminating in the proof
of ergodicity of four billiard balls in the torus $\Bbb T^\nu$
($\nu\ge 3$), [K-S-Sz(1992)]. Then, in 1992, Bunimovich, Liverani,
Pellegrinotti and Sukhov [B-L-P-S(1992)] were able to prove the ergodicity for
some systems with an arbitrarily 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
additional
scattering effect with the collisions at the strictly concave wall of the
container. The only result with an arbitrarily large number of balls in a
flat unit torus $\Bbb T^\nu$ was achieved in the twin papers of mine
[Sim(1992-I-II)], 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.
On the other hand, if someone considers a hard ball system in an
elongated torus which is long in one direction but narrow in the others,
so that the balls must keep their cyclic order in the ``long direction''
(Sinai's ``pen-case'' model), then the technical difficulties can be handled,
thanks to the fact that the collisions of balls are now restricted to
neighboring pairs. The hyperbolicity of such models in three dimensions and
the ergodicity in dimension four have been proved in [S-Sz(1995)].
The positivity of the metric entropy for several systems of hard balls
can be proven relatively easily, as was shown in the paper [W(1988)].
The articles [L-W(1995)] and [W(1990)] are nice surveys describing a
general setup leading to the technical problems treated in a series of
research papers. For a comprehensive survey of the results and open problems
in this field, see [Sz(1996)].
Pesin's theory [P(1977)] on the ergodic properties of non-uniformly hyperbolic,
smooth dynamical systems has been generalized substantially to dynamical
systems with singularities (and with a relatively mild behavior near the
singularities) by A. Katok and J-M. Strelcyn [K-S(1986)]. Since then, the
so called Pesin's and Katok-Strelcyn's theories have become part of the
folklore in the theory of dynamical systems. They claim that --- under some
mild regularity conditions, particularly near the singularities --- every
non-uniformly hyperbolic and ergodic flow enjoys the Kolmogorov-mixing
property, shortly the K-mixing property.
Later on it was discovered and proven in [C-H(1996)] and [O-W(1998)] that the
above mentioned fully hyperbolic and ergodic flows with singularities turn out
to be automatically having the Bernoulli mixing (B-mixing) property. It is
worth noting here that almost every semi-dispersive billiard system,
especially every hard ball system, enjoys those mild regularity conditions
imposed on the systems (as axioms) by [K-S(1986)], [C-H(1996)], and
[O-W(1998)]. In other words, for a hard ball flow
$\left(\bold M,\{S^t\},\mu\right)$ the (global) ergodicity of the system
actually implies its full hyperbolicity and the B-mixing property, as well.
Finally, in our joint venture with D. Sz\'asz [S-Sz(1999)], 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 complete hyperbolicity (nonzero Lyapunov
exponents almost everywhere) for $N$ balls in $\Bbb T^\nu$. The ergodicity
appeared to be a harder task.
We note, however, that the algebraic method developed in [S-Sz(1999)]
is being further developed in this paper in order to obtain ergodicity,
not only full hyperbolicity.
Consider the $\nu$-dimensional ($\nu\ge2$), standard, flat torus
$\Bbb T_L^\nu=\Bbb R^\nu/L\cdot\Bbb Z^\nu$ 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 $\bold Q$ 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
$$
C_{i,j}=\left\{(q_1,\dots,q_N)\in\Bbb T^{\nu(N-1)}\colon\;
\text{dist}(q_i,q_j)<2r \right\}
$$
($1\le i0$, $L>L_0(r,\,\nu)$ where the interior
of the phase space is connected it is true that the billiard flow
$\left(\bold M_{\vec m,L},\{S^t\},\mu_{\vec m,L}\right)$ of the $N$-ball
system is ergodic and completely hyperbolic. Then, following from the results
of Chernov--Haskell [C-H(1996)] and Ornstein--Weiss [O-W(1998)], such a
semi-dispersive billiard system actually enjoys the B-mixing property, as well.
\medskip
\subheading{Remark 1} We note that the main result
of this paper and that of [Sim(2001)] nicely complement each other. They
precisely assert the same, almost sure ergodicity of hard ball systems in the
cases $\nu\ge3$ and $\nu=2$, respectively. It should be noted, however, that
the proof of [Sim(2001)] is primarily dynamical--geometric (except the
verification of the Chernov-Sinai Ansatz), whereas the novel parts of the
present proof are fundamentally algebraic.
\medskip
\subheading{Remark 2} The above inequality $L>L_0(r,\,\nu)$ corresponds to
physically relevant situations. Indeed, in the case $L0$ (the radii of the spherical cylinders $C_i$) and some translation
vectors $t_i\in\Bbb T^d=\Bbb R^d/\Cal L$ be given. The translation
vectors $t_i$ play a role in positioning the cylinders $C_i$
in the ambient torus $\Bbb T^d$. Set
$$
C_i=\left\{x\in\Bbb T^d\colon\; \text{dist}\left(x-t_i,A_i/(A_i\cap\Cal L)
\right)L_0(r,\,\nu)$ where (2.1.1) is true. Therefore, in the
sense of our theorem of ``almost sure ergodicity'', the non-degeneracy condition
(2.1.2) does not mean a restriction of generality.
\bigskip
\subheading{2.2 Hard ball systems} Hard ball systems in the flat
torus $\Bbb T^\nu_L=\Bbb R^\nu/L\cdot\Bbb Z^\nu$ ($\nu\ge2$) with positive masses
$m_1,\dots,m_N$ are described (for example) in \S 1 of [S-Sz(1999)].
These are the dynamical systems describing the motion of $N$ ($\ge2$) hard
balls with a common radius $r>0$ and positive masses $m_1,\dots,m_N$ in
the flat torus of size $L$, $\Bbb T^\nu_L=\Bbb R^\nu/L\cdot\Bbb Z^\nu$.
(Just for simplicity, we will assume that the radii have the common
value $r$.) The center of the
$i$-th ball is denoted by $q_i$ ($\in\Bbb T^\nu_L$), its time derivative is
$v_i=\dot q_i$, $i=1,\dots,N$. One uses the standard reduction of kinetic
energy $E=\frac{1}{2}\sum_{i=1}^N m_i||v_i||^2=\frac{1}{2}$.
The arising configuration space (still without the removal of the scattering
cylinders $C_{i,j}$) is the torus
$$
\Bbb T_L^{\nu N}=\left(\Bbb T_L^{\nu}\right)^N=\left\{(q_1,\dots,q_N)\colon\;
q_i\in\Bbb T_L^\nu,\; i=1,\dots,N\right\}
$$
supplied with the Riemannian inner product (the so called mass metric)
$$
\langle v,v'\rangle=\sum_{i=1}^N m_i\langle v_i,v'_i \rangle
\tag 2.2.1
$$
in its common tangent space $\Bbb R^{\nu N}=\left(\Bbb R^{\nu}\right)^N$.
Now the Euclidean space $\Bbb R^{\nu N}$ with the inner product (2.2.1)
plays the role of $\Bbb R^d$ in the original definition of cylindric
billiards, see \S 2.1 above.
The generator subspace $A_{i,j}\subset \Bbb R^{\nu N}$ ($1\le i0) \;
\text{ such that } \; \forall \alpha \in (-\delta,\delta) \\
&V\left(S^a\left(Q(x)+\alpha W,V(x)\right)\right)=V(S^ax)\text{ and }
V\left(S^b\left(Q(x)+\alpha W,V(x)\right)\right)=V(S^bx)\big\}.
\endaligned
$$
\endproclaim \endrem
\noindent
($\Cal Z$ is the common tangent space $\Cal T_q\bold Q$ of the parallelizable
manifold $\bold Q$ at any of its points $q$, while $V(x)$ is the velocity
component of the phase point $x=\left(Q(x),\,V(x)\right)$.)
It is known (see (3) in \S 3 of [S-Ch (1987)]) that
$\Cal N_0(S^{[a,b]}x)$ is a linear subspace of $\Cal Z$ 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, \S 2 of [K-S-Sz(1990)]).
Our next definition is that of the advance. Consider a
non-singular orbit segment $S^{[a,b]}x$ with the symbolic collision sequence
$\Sigma=(\sigma_1, \dots, \sigma_n)$ ($n\ge 1$), meaning that $S^{[a,b]}x$
has exactly $n$ collisions with $\partial\bold Q$, and the $i$-th collision
($1\le i\le n$) takes place at the boundary of the cylinder $C_{\sigma_i}$.
For $x=(Q,V)\in\bold M$ and $W\in\Cal Z$, $\Vert W\Vert$ sufficiently small,
denote $T_W(Q,V):=(Q+W,V)$.
\proclaim{Definition 2.5.2}
For any $1\le k\le n$ and $t\in[a,b]$, the advance
$$
\alpha(\sigma_k)\colon\;\Cal N_t(S^{[a,b]}x) \rightarrow \Bbb R
$$
of the collision $\sigma_k$ is the unique linear extension of the linear
functional $\alpha(\sigma_k)$
defined in a sufficiently small neighborhood of the origin of
$\Cal N_t(S^{[a,b]}x)$ in the following way:
$$
\alpha(\sigma_k)(W):= t_k(x)-t_k(S^{-t}T_WS^tx).
$$
\endproclaim \endrem
Here $t_k=t_k(x)$ is the time moment of the $k$-th collision $\sigma_k$ on
the trajectory of $x$ after time $t=a$. The above formula and the notion of
the advance functional
$$
\alpha_k=\alpha(\sigma_k):\; \Cal N_t\left(S^{[a,b]}x\right)\to\Bbb R
$$
has two important features:
\medskip
(i) If the spatial translation $(Q,V)\mapsto(Q+W,V)$ is carried out at time
$t$, then $t_k$ changes linearly in $W$, and it takes place just
$\alpha_k(W)$ units of time earlier. (This is why it is called ``advance''.)
\medskip
(ii) If the considered reference time $t$ is somewhere between $t_{k-1}$
and $t_k$, then the neutrality of $W$ with respect to $\sigma_k$ precisely
means that
$$
W-\alpha_k(W)\cdot V(x)\in A_{\sigma_k},
$$
i. e. a neutral (with respect to the collision $\sigma_k$) spatial translation
$W$ with the advance $\alpha_k(W)=0$ means that the vector $W$ belongs to the
generator space $A_{\sigma_k}$ of the cylinder $C_{\sigma_k}$.
It is now time to bring up the basic notion of sufficiency
(or, sometimes it is also called geometric hyperbolicity) of a
trajectory (segment). This is the utmost important necessary condition for
the proof of the fundamental theorem for algebraic semi-dispersive billiards,
see Theorem 4.4 in [B-Ch-Sz-T(2001)].
\medskip
\proclaim{Definition 2.5.3}
\roster
\item
The nonsingular trajectory segment $S^{[a,b]}x$ ($a$ and $b$ are supposed not
to be moments of collision) is said to be sufficient if and only if
the dimension of $\Cal N_t(S^{[a,b]}x)$ ($t\in [a,b]$) is minimal, i.e.
$\text{dim}\ \Cal N_t(S^{[a,b]}x)=1$.
\item
The trajectory segment $S^{[a,b]}x$ containing exactly one singularity (a so
called ``simple singularity'', see 2.4 above) is said to be sufficient if
and only if both branches of this trajectory segment are sufficient.
\endroster
\endproclaim \endrem
\medskip
\proclaim{Definition 2.5.4}
The phase point $x\in\bold M$ with at most one (simple) singularity is said
to be sufficient if and only if its whole trajectory $S^{(-\infty,\infty)}x$
is sufficient, which means, by definition, that some of its bounded
segments $S^{[a,b]}x$ are sufficient.
\endproclaim \endrem
In the case of an orbit $S^{(-\infty,\infty)}x$ with a simple
singularity, sufficiency means that both branches of
$S^{(-\infty,\infty)}x$ are sufficient.
\bigskip
\subheading{2.6. No accumulation (of collisions) in finite time}
By the results of Vaserstein [V(1979)], Galperin [G(1981)] and
Burago-Ferleger-Kononenko [B-F-K(1998)], in a semi-dis\-per\-sive billiard
flow with the property (2.1.2) there can only be finitely many
collisions in finite time intervals, see Theorem 1 in [B-F-K(1998)].
Thus, the dynamics is well defined as long as the trajectory does not hit
more than one boundary components at the same time.
\bigskip
\subheading{2.7. Slim sets}
We are going to summarize the basic properties of codimension-two subsets $A$
of a connected, smooth manifold $M$ with a possible boundary. Since these subsets
$A$ are just those negligible in our dynamical discussions, we shall call them
slim. As to a broader exposition of the issues, see [E(1978)] or \S2 of
[K-S-Sz(1991)].
Note that the dimension $\dim A$ of a separable metric space $A$ is one of the
three classical notions of topological dimension: the covering
(\v Cech-Lebesgue), the small inductive (Menger-Urysohn), or the large
inductive (Brouwer-\v Cech) dimension. As it is known from general general
topology, all of them are the same for separable metric spaces.
\medskip
\subheading{Definition 2.7.1}
A subset $A$ of $M$ is called slim if and only if $A$ can be covered by a
countable family of codimension-two (i. e. at least two) closed sets of
$\mu$--measure zero, where $\mu$ is a smooth measure on $M$. (Cf.
Definition 2.12 of [K-S-Sz(1991)].)
\medskip
\subheading{Property 2.7.2} The collection of all slim subsets of $M$ is a
$\sigma$-ideal, that is, countable unions of slim sets and arbitrary
subsets of slim sets are also slim.
\medskip
\subheading{Proposition 2.7.3. (Locality)}
A subset $A\subset M$ is slim if and only if for
every $x\in A$ there exists an open neighborhood $U$ of $x$ in $M$ such that
$U\cap A$ is slim. (Cf. Lemma 2.14 of [K-S-Sz(1991)].)
\medskip
\subheading{Property 2.7.4} A closed subset $A\subset M$ is slim if and only
if $\mu(A)=0$ and $\dim A\le\dim M-2$.
\medskip
\subheading{Property 2.7.5. (Integrability)}
If $A\subset M_1\times M_2$ is a closed subset of the product of two smooth
manifolds with possible boundaries, and for every $x\in M_1$ the set
$$
A_x=\{ y\in M_2\colon\; (x,y)\in A\}
$$
is slim in $M_2$, then $A$ is slim in $M_1\times M_2$.
\medskip
The following propositions characterize the codimension-one and
codimension-two sets.
\medskip
\subheading{Proposition 2.7.6}
For any closed subset $S\subset M$ the following three conditions are
equivalent:
\roster
\item"{(i)}" $\dim S\le\dim M-2$;
\item"{(ii)}" $\text{int}S=\emptyset$ and for every open connected set
$G\subset M$ the difference set $G\setminus S$ is also connected;
\item"{(iii)}" $\text{int}S=\emptyset$ and for every point $x\in M$ and for any
open neighborhood $V$ of $x$ in $M$ there exists a smaller open neighborhood
$W\subset V$ of the point $x$ such that for every pair of points
$y,z\in W\setminus S$ there is a continuous curve $\gamma$ in the set
$V\setminus S$ connecting the points $y$ and $z$.
\endroster
\noindent
(See Theorem 1.8.13 and Problem 1.8.E of [E(1978)].)
\medskip
\subheading{Proposition 2.7.7} For any subset $S\subset M$ the condition
$\dim S\le\dim M-1$ is equivalent to $\text{int}S=\emptyset$.
(See Theorem 1.8.10 of [E(1978)].)
\medskip
We recall an elementary, but important lemma (Lemma 4.15 of [K-S-Sz(1991)]).
Let $R_2$ be the set of phase points
$x\in\bold M\setminus\partial\bold M$ such that the trajectory
$S^{(-\infty,\infty)}x$ has more than one singularities.
\subheading{Proposition 2.7.8}
The set $R_2$ is a countable union of codimension-two
smooth sub-manifolds of $M$ and, being such, it is slim.
\medskip
The next lemma establishes the most important property of slim sets which
gives us the fundamental geometric tool to connect the open ergodic components
of billiard flows.
\medskip
\subheading{Proposition 2.7.9}
If $M$ is connected, then the complement $M\setminus A$ of a slim $F_\sigma$
set $A\subset M$ is an arc-wise connected ($G_\delta$) set of full measure.
(See Property 3 of \S 4.1 in [K-S-Sz(1989)]. The $F_\sigma$ sets are,
by definition, the countable unions of closed sets, while the $G_\delta$ sets
are the countable intersections of open sets.)
\medskip
\subheading{2.8. The subsets $\bold M^0$ and $\bold M^\#$} Denote by
$\bold M^\#$ the set of all phase points $x\in\bold M$ for which the
trajectory of $x$ encounters infinitely many non-tangential collisions
in both time directions. The trajectories of the points
$x\in\bold M\setminus\bold M^\#$ are lines: the motion is linear and uniform,
see the appendix of [Sz(1994)]. It is proven in lemmas A.2.1 and A.2.2
of [Sz(1994)] that the closed set $\bold M\setminus\bold M^\#$ is a finite
union of hyperplanes. It is also proven in [Sz(1994)] that, locally, the two
sides of a hyper-planar component of $\bold M\setminus\bold M^\#$ can be
connected by a positively measured beam of trajectories, hence, from the point
of view of ergodicity, in this paper it is enough to show that the connected
components of $\bold M^\#$ entirely belong to one ergodic component. This is
what we are going to do in this paper.
Denote by $\bold M^0$ the set of all phase points $x\in\bold M^\#$ the
trajectory of which does not hit any singularity, and use the notation
$\bold M^1$ for the set of all phase points $x\in\bold M^\#$ whose orbit
contains exactly one, simple singularity. According to Proposition 2.7.8,
the set $\bold M^\#\setminus(\bold M^0\cup\bold M^1)$ is a countable union of
smooth, codimension-two ($\ge2$) submanifolds of $\bold M$, and, therefore,
this set may be discarded in our study of ergodicity, please see also the
properties of slim sets above. Thus, we will restrict our attention to the
phase points $x\in\bold M^0\cup\bold M^1$.
\medskip
\subheading{2.9. The ``Chernov-Sinai Ansatz''} An essential precondition for
the Theorem on Local Ergodicity by B\'alint--Chernov--Sz\'asz--T\'oth
(Theorem 4.4 of [B-Ch-Sz-T(2001)]) is the
so called ``Chernov-Sinai Ansatz'' which we are going to formulate below.
Denote by $\Cal S\Cal R^+\subset\partial\bold M$ the set of all phase points
$x_0=(q_0,v_0)\in\partial\bold M$ corresponding to singular reflections
(a tangential or a double collision at time zero) supplied with the
post-collision (outgoing) velocity $v_0$. It is well known that
$\Cal S\Cal R^+$ is a compact cell complex with dimension
$2d-3=\text{dim}\bold M-2$. It is also known (see Lemma 4.1 in [K-S-Sz(1990)])
that for $\nu$-almost every phase point $x_0\in\Cal S\Cal R^+$ the forward
orbit $S^{(0,\infty)}x_0$ does not hit any further singularity.
(Here $\nu$ is the Riemannian volume of $\Cal S\Cal R^+$ induced by the
restriction of the natural Riemannian metric of $\bold M$.)
The Chernov-Sinai Ansatz postulates that for $\nu$-almost every
$x_0\in\Cal S\Cal R^+$ the forward orbit $S^{(0,\infty)}x_0$ is sufficient
(geometrically hyperbolic).
\medskip
\subheading{2.10. The Theorem on Local Ergodicity} The Theorem on Local
Ergodicity by B\'alint--Chernov--Sz\'asz--T\'oth (Theorem 4.4 of
[B-Ch-Sz-T(2001)]) claims the following: Let $\flow$ be a semi-dispersive
billiard flow with (2.1.1)--(2.1.2) and with the property
that the smooth components of the boundary $\partial\bold Q$ of the
configuration space are algebraic hyper-surfaces. (The cylindric billiards
automatically fulfill this algebraicity condition.) Assume -- further --
that the Chernov-Sinai Ansatz holds true, and a phase point
$x_0\in\left(\bold M\setminus\partial\bold M\right)\cap\bold M^\#$ is
given with the properties
\medskip
(i) $S^{(-\infty,\infty)}x$ has at most one singularity,
\noindent
and
(ii) $S^{(-\infty,\infty)}x$ is sufficient.
\medskip
Then some open neighborhood $U_0\subset\bold M$ of $x_0$ belongs to a single
ergodic component of the flow $\flow$. (Modulo the zero sets, of course.)
\bigskip \bigskip
\heading
3. The Algebraic Approach
\endheading
\bigskip \bigskip
In this paper we would like to present a proof for the Theorem as
self-contained as possible. For that reason, below we are inserting \S3
and the first part of \S4 of the paper [S-Sz(1999)]. Only small editions
have been performed on the original text in order to make it fully
compatible with the present article. At the end of this section we cite
all other results proved in [S-Sz(1999)] that are deemed to be necessary for
the present proof.
The aim of this section is to properly understand the algebraic relationship
between the kinetic data of the billiard flow measured at different moments
of time.
From now on we shall investigate orbit segments
$S^{[0,T]}x_0$ ($T>0$) of the standard billiard ball flow
$\left(\bold M,\{S^t\},\mu,\vec m,L \right)$. (Here $L>0$ is the size of
the container torus $\Bbb T_L^\nu=\Bbb R^\nu/\left(L\cdot\Bbb Z^\nu\right)$
in which the balls of radius $r$ are moving.) We note here that
$\bold M=\bold Q\times\Bbb S^{d-1}$, where the configuration space
$\bold Q$ was defined in \S2 and $\Bbb S^{d-1}$ is the velocity sphere
$$
\Bbb S^{d-1}=\left\{(v_1,\dots,v_N)\in\Bbb R^{\nu N}\bigg|\,\sum_{i=1}^N m_iv_i=0
\; \& \; \sum_{i=1}^N m_i\Vert v_i\Vert^2=1 \right\}
$$
introduced in \S2. Also note that in the geometric-algebraic
considerations of the upcoming sections we do not use the factorization with
respect to uniform spatial translations introduced in \S2.
Later on even the conditions
$\sum_{i=1}^N m_iv_i=0$ and $\sum_{i=1}^N m_i\Vert v_i\Vert^2=1$ will be
dropped, cf. Remark 3.14 below.
The symbolic collision sequence of $S^{[0,T]}x_0$ is denoted by
$\Sigma\left(S^{[0,T]}x_0\right)=(\sigma_1,\sigma_2,\allowmathbreak\dots ,\sigma_n)$.
The symbol
$v_i^k=v_i^+(t_k)\in\Bbb R^{\nu}$ shall denote the velocity $\dot q_i$ of
the $i$-th ball right {\it after} the $k$-th collision $\sigma_k=\{i_k,j_k\}$
($1\le i_k0$ of the torus and, finally,
the masses $m_i$ as constants.
Since these algebraic functions make full sense over the complex field
$\Bbb C$ and, after all, our proof of the theorem requires the
complexification, we are now going to complexify the whole system by
considering the kinetic variables, the size $L$, and the masses as complex
ones and retaining the polynomial equations (3.4)-(3.8). The fixed radius
$r>0$ plays the role of a constant, not a variable. However, due to the ambiguity of
selecting a root of (3.8) out of the two, it proves to be important to explore
first the algebraic framework of the relations (3.4)-(3.8) connecting the studied
variables. This is what we do now.
\bigskip
\heading
The Field Extension Associated With the Pair $\left(\Sigma,\Cal A\right)$
\endheading
\medskip
To avoid misunderstanding, we immediately stress that the field extensions
$\Bbb K_n=\Bbb K_n\left(\Sigma;\Cal A\right)$ to be defined below will also
depend on a sequence $\vec\tau=(\tau_1, \dots, \tau_n)$ of field elements
to be introduced iteratively in Definition 3.11 below. If we also want to
emphasize the dependence of $\Bbb K$ on $\vec\tau$, then we will write
$\Bbb K_n=\Bbb K_n\left(\Sigma;\Cal A\right)=\Bbb K_n\harmas$.
We are going to define the function field
$\Bbb K_n=\Bbb K_n\left(\Sigma;\Cal A\right)$ generated by all functions
$$
\left\{(\tilde q_i^k)_j,\, (v_i^k)_j|\, i=1,\dots,N;\, k=0,\dots,n;\,
j=1,\dots,\nu\right\}
$$
of the lifted orbit segments corresponding to the fixed parameters
$$
\left(\Sigma,\Cal A\right)=(\sigma_1,\dots,\sigma_n; a_1,\dots,a_n)
$$
in such a way that the field $\Bbb K_n\left(\Sigma;\Cal A\right)$ incorporates
all algebraic relations among these variables that are consequences of the
equations (3.4)-(3.8). (Here the subscript $j$ denotes the $j$-th component
of a $\nu$-vector.) In our setup the ground field (the coefficient field)
of allowed coefficients is, by definition, the complex field $\Bbb C$.
The precise definition of $\Bbb K_n=\Bbb K_n\left(\Sigma;\Cal A\right)$ is
\medskip
\subheading{Definition 3.11} For $n=0$ the field $\Bbb K_0=\Bbb K_0(\emptyset;
\emptyset)$ is the transcendental extension $\Bbb C(\Cal B)$ of the
coefficient field $\Bbb C$ by the algebraically independent formal initial
variables
$$
\Cal B=\left\{(\tilde q_i^0)_j,\, (v_i^0)_j,\, m_i,\, L|\, i=1,\dots,N;\,
j=1,\dots,\nu \right\}.
$$
Suppose now that $n>0$ and the commutative field
$\Bbb K_{n-1}=\Bbb K_{n-1}\left(\Sigma';\Cal A'\right)$ has already been defined,
where $\Sigma'=(\sigma_1,\dots,\sigma_{n-1})$, $\Cal A'=(a_1,\dots,a_{n-1})$.
Then we consider the quadratic equation $b_n\tau_n^2+c_n\tau_n+d_n=0$
in (3.8) with $k=n$ as a polynomial equation defining a new field element
$\tau_n$ to be adjoined to the field
$\Bbb K_{n-1}=\Bbb K_{n-1}\left(\Sigma';\Cal A'\right)$. (Recall that the
coefficients $b_n$, $c_n$, $d_n$ come from the field $\Bbb K_{n-1}$.)
There are two possibilities:
(i) The quadratic polynomial $b_nx^2+c_nx+d_n$ is reducible over the field
$\Bbb K_{n-1}$. Then we take $\Bbb K_n=\Bbb K_{n-1}$.
(ii) The polynomial $b_nx^2+c_nx+d_n$ is irreducible over the field
$\Bbb K_{n-1}$. Then we define $\Bbb K_n=\Bbb K_n\left(\Sigma;\Cal A\right)$
as the extension of $\Bbb K_{n-1}=\Bbb K_{n-1}\left(\Sigma';\Cal A'\right)$
by the root $\tau_n$ of this irreducible polynomial.
\medskip
\subheading{Remark 3.12} The importance of the field $\Bbb K_n$ is
underscored by the fact that this field encodes all algebraic relations among
the kinetic data that follow from the polynomial equations (3.4)-(3.8).
Furthermore, Proposition 3.3 gives us an iterative computation rule for
successively obtaining the kinetic variables
$$
\left\{\tilde q_i^k,\, v_i^k|\, i=1,\dots,N\right\},
$$
$k=0,\dots,n$.
The field $\Bbb K_n=\Bbb K_n(\Sigma,\Cal A)$ is, after all, the algebraic framework
of such computations. We note that -- everywhere in this paper -- the
symbols $\langle\,.\,;\,.\,\rangle$ and $\Vert\,.\,\Vert^2$ do
not refer to an Hermitian
inner product but, rather, $\langle x;y \rangle=\sum_{j=1}^{\nu}x_jy_j$ and
$\Vert x\Vert^2=\sum_{j=1}^{\nu}x_j^2$, so that these expressions retain their
polynomial form.
\subheading{Remark 3.13} Later on it will be necessary to express each
kinetic variable $\left(\tilde q_i^k\right)_j$ and $\left(v_i^k\right)_j$
as an algebraic function of the initial variables $\Cal B$. This raises,
however, an important question: Which one of the two roots $\tau_k$ of
(3.8) should be considered during these computations? This is no problem if
the polynomial $b_kx^2+c_kx+d_k$, defining $\tau_k$, is irreducible over the
field $\Bbb K_{k-1}$, because the two roots of this polynomial are
algebraically equivalent over $\Bbb K_{k-1}$, and any of them can be chosen
as $\tau_k$. However, if the polynomial $b_kx^2+c_kx+d_k$
is reducible over the field $\Bbb K_{k-1}$, then it is necessary to make a
decision and assign to one of the two roots (both in the field $\Bbb K_{k-1}$
now) the role of $\tau_k$. This is what we do. The result is the field
$\Bbb K_n=\Bbb K_n(\Sigma;\Cal A)$ endowed with a distinguished
n-tuple $(\tau_1, \dots, \tau_n)$ of its elements. The field $\Bbb K_n$ with
the distinguished $n$-tuple $\vec\tau$ is denoted by
$\Bbb K_n=\Bbb K_n(\Sigma;\Cal A;\vec\tau)$. Then the algebraic object
$\Bbb K_n(\Sigma;\Cal A;\vec\tau)$ completely controls the whole process of the
iterative computation of the kinetic variables.
Note that in the genuine, real case the two roots of $b_kx^2+c_kx+d_k$ are
distinct, positive real numbers, and -- by the nature of the billiard
dynamics -- the chosen value is always the smaller one.
Finally, we mention here a simple case when the polynomial
$b_kx^2+c_kx+d_k$ is reducible over the field $\Bbb K_{k-1}$. Namely, this
takes place whenever $\sigma_{k-1}=\sigma_k$ and $a_{k-1}=a_k$. In that case
the value $\tau_k=0$ is clearly a solution of the above polynomial
$b_kx^2+c_kx+d_k$, and, being so, that polynomial is reducible over the field
$\Bbb K_{k-1}$, see also 3.31--3.32 below.
\subheading{Remark 3.14} The reader may wonder why we did not postulate
the algebraic dependencies $\sum_{i=1}^N m_iv_i^k=0$,
$\sum_{i=1}^N m_i\Vert v_i^k\Vert^2=1$? The answer is the following:
The definition of the neutral linear space together with its characterization
via the Connecting Path Formula (CPF, see Lemma 2.9 of [Sim(1992)-II] or
Proposition 2.19 in [S-Sz(1999)]), i. e. the partial linearity of the
billiard flow, are invariant under
\medskip
(1) all uniform velocity translations
(adding the same vector to all velocities) and
(2) time rescaling.
\subheading{Remark 3.15} All fields occurring in this paper are only defined
up to an isomorphism over the coefficient field $\Bbb C$. Therefore, the
statements like ``the field $\Bbb K_2$ is an extension of $\Bbb K_1$''
should be understood as follows: $\Bbb K_1$ is a subfield of $\Bbb K_2$
after the natural identification of the generating variables bearing the
same name. For the necessary notions, properties, and results from
the theory of field extensions and Galois theory, the reader is kindly
recommended to look up, for instance, the book by I. Stewart, [St(1973)].
\medskip
\subheading{Remark 3.16} As said before (see Definition 3.11), the
collection $\Cal B$ of elements of the field
$\Bbb K_n(\Sigma;\Cal A; \vec\tau)$
is a base of transcendence in that field over the subfield $\Bbb C$, and the
degree of the extension
$\left|\Bbb K_n(\Sigma;\Cal A;\vec\tau):\Bbb C(\Cal B)\right|=\left|\Bbb K_n:\Bbb K_0\right|$
is a power of two. Moreover, each of the following sets is a generator for the field
$\Bbb K_n$:
\medskip
(a) $\Bbb C\cup\{m_1,\dots,m_N,L\}\cup\left\{(\tilde q_i^0)_j,\,
(v_i^0)_j|\, i=1,\dots,N;\, j=1,\dots,\nu\right\}\cup\{\tau_1,\dots,\tau_n\}$;
(b) $\Bbb C\cup\{m_1,\dots,m_N,L\}\cup\left\{(\tilde q_i^k)_j,\,
(v_i^k)_j|\, i=1,\dots,N;\, j=1,\dots,\nu;\, k=0,\dots,n\right\}$.
\subheading{Remark 3.17} The above procedure of constructing field
extensions is closely related to the classical theory of geometric constructions
by a straight edge and a compass, and this is not for
surprise: The billiard trajectory is constructed by intersecting a straight
line with a sphere and then reflecting it across the tangent hyperplane of the
sphere. This is a sort of classical geometric construction in $\nu$ dimensions.
For the details see, for instance, \S57 and \S60 of [VDW (1970)], or
Chapter 5 of [St(1973)].
Let us fix the pair $(\Sigma;\Cal A)$ and the $n$-tuple
$\vec\tau=(\tau_1,\dots,\tau_n)$ of elements of the field
$\Bbb K_n(\Sigma, \Cal A)$.
We are now defining a $(2\nu+1)N+1$-dimensional
complex analytic manifold $\Omega=\Omega\harmas$ and certain
holomorphic functions $\tilde q_i^k,\, v_i^k:\,\Omega\to\Bbb C^{\nu}$,
$m_i:\,\Omega\to\Bbb C$, $L:\,\Omega\to\Bbb C$,
$\tau_k:\,\Omega\to\Bbb C$,
$i=1,\dots,N$, $k=0,\dots,n$, ($\tau_0$ is not defined).
But first we introduce
\medskip
\subheading{Definition 3.18} Define the domain
$D=D\harmas\subset\Bbb C^{(2\nu+1)N+1}$ as the set of all complex
$(2\nu+1)N+1$-tuples
$$
\left\{(\tilde q_i^0)_j,\, (v_i^0)_j,\, m_i,\, L|\, i=1,\dots,N;\,
j=1,\dots,\nu \right\}
$$
for which
\medskip
(a) the leading coefficient
$\left\Vert v_{i_k}^{k-1}-v_{j_k}^{k-1}\right\Vert^2$ and the discriminant
of the quadratic equation (3.8) is never zero, $k=1,\dots,n$,
(The latter condition is equivalent to
$$
\left\langle v_{i_k}^{k-1}-v_{j_k}^{k-1};\,
\tilde q_{i_k}^k-\tilde q_{j_k}^k-L\cdot a_k\right\rangle\ne 0,
$$
see also Remark 3.29), and
\medskip
(b) $m_{i_k}+m_{j_k}\ne 0$ for $k=1,\dots,n$,
\noindent
provided that we carry out the iterative computation of the kinetic variables
$$
\left\{(\tilde q_i^k)_j,\, (v_i^k)_j|\, i=1,\dots,N;\, j=1,\dots,\nu;\,
k=0,\dots,n\right\}
$$
as governed by the polynomial equations (3.4)-(3.8) and by the fixed selection
of elements $\vec\tau$.
\medskip
\subheading{Remark 1} When inverting the dynamics, a useful remark is that
$$
\left\Vert v_{i_k}^{k-1}-v_{j_k}^{k-1}\right\Vert^2
=\left\Vert v_{i_k}^{k}-v_{j_k}^{k}\right\Vert^2,
$$
and
$$
\left\langle v_{i_k}^{k-1}-v_{j_k}^{k-1};\,
\tilde q_{i_k}^k-\tilde q_{j_k}^k-L\cdot a_k\right\rangle =
-\left\langle v_{i_k}^{k}-v_{j_k}^{k};\,
\tilde q_{i_k}^k-\tilde q_{j_k}^k-L\cdot a_k\right\rangle.
$$
\medskip
\subheading{Remark 2} Note that we require the validity of (a) (of 3.18)
for any branch of the square root function, when computing the $\tau_k$'s
with irreducible defining polynomials (3.8).
\subheading{Lemma 3.19}
The complement set $\Bbb C^{(2\nu+1)N+1}\setminus D$ is a proper,
closed, algebraic subset of $\Bbb C^{(2\nu+1)N+1}$, especially a finite union
of complex analytic submanifolds with codimension at least one, so the open
set $D\subset\Bbb C^{(2\nu+1)N+1}$ is {\it connected and dense}.
\medskip
\subheading{Proof}
We have the polynomial equations (3.4)-(3.8) making it
possible to set up an algorithm for iteratively computing the kinetic
variables measured at different times. The point is not just this iterative
algorithm, but also its invertibility (time reversibility). The inverse
process has similar algebraic properties, for it just means time reversal.
One can easily prove by an induction on the length $n$ of $\Sigma$ that the
iteratively defined dynamics determines a several-to-several mapping with
maximum rank between a nonempty Zariski open set of the kinetic
variables with superscript $1$ and a nonempty Zariski open set of the
kinetic variables measured right after the last reflection
(with superscript $n$). (A Zariski open set is the complement of a closed
algebraic set defined as the simultaneous zero set of finitely many
polynomials.) Then one can go ahead with the induction from $n$ to $n+1$,
because all possible obstructions to extending the process only occur on
proper algebraic submanifolds of the Zariski open set of kinetic variables
with superscript $n$. \qed
\subheading{Definition 3.20} We define $\Omega\harmas$ as the set of all
complex $(2\nu(n+1)N+N+n+1)$-tuples
$$
\aligned
& \omega= \\
& \bigg(\left(\tilde q_i^k\right)_j,\, (v_i^k)_j,\, m_i,\, L,\,
\tau_l \bigg| \, i=1,\dots,N;\, k=0,\dots,n;\, j=1,\dots,\nu;\,
l=1,\dots,n \bigg)
\endaligned
$$
for which these coordinates
\roster
\item
are interrelated by the equations (3.4)--(3.8),
\item
respect the choices of $\tau_k$ prescribed by $\vec\tau$
(cf 3.11, and also 3.13),
\endroster
\noindent
and the vector
$$
\vec x(\omega)=\left(\left(\tilde q_i^0\right)_j,\, (v_i^0)_j,\, m_i,\, L
\big|\, i=1,\dots,N;\, j=1,\dots,\nu \right)
$$
of initial data belongs to the set $D\harmas$ defined above.
\medskip
It is clear now that $\tilde q_i^k,\, v_i^k:\,\Omega\to\Bbb C^{\nu}$,
$m_i:\,\Omega\to\Bbb C$, $L:\,\Omega\to\Bbb C$,
and $\tau_k:\,\Omega\to\Bbb C$
($i=1,\dots,N$, $k=0,\dots,n$) are holomorphic functions on
$\Omega=\Omega\harmas$. The complex analytic manifold
$\Omega$ endowed with the above holomorphic functions can justifiably
considered as the complexification of the $\harmas$-iterated billiard map.
\subheading{Remark} The careful reader has certainly done the following
observation: The $\tau_k$'s figure in two different roles.
They first denote successively chosen field elements of the extensions
introduced in Definition 3.11, and secondly they also denote complex-valued
functions on $\Omega$ (or multi-valued complex functions on $D$, cf.
Definitions 3.20 and 3.18).
\subheading{Remark} It is very likely that the manifold $\Omega$
is connected. Nevertheless, in our proof we do not need this connectedness,
and are not going to pursue the goal of proving it.
\bigskip
\heading
The Complex Neutral Space $\Cal N(\omega)$
\endheading
\medskip
Fix a base point $\omega\in\Omega\harmas$. The tangent space
$\Cal T_{\omega}\Omega$ of $\Omega$ at $\omega$ consists of all
complex vectors
$$
x=\left(\delta\tilde q_i^0,\, \delta v_i^0,\, \delta m_i,\, \delta L\,
|\, i=1,\dots,N\right)\in\Bbb C^{(2\nu+1)N+1},
$$
thus $\Cal T_{\omega}\Omega$ can be naturally identified with the
complex vector space $\Bbb C^{(2\nu+1)N+1}$. Set
$$
\aligned
\neutr=\bigg\{x=\left(\delta\tilde q_i^0,\, \delta v_i^0,\,
\delta m_i,\, \delta L\, |\,
i=1,\dots,N \right)\in\Cal T_{\omega}\Omega\bigg| \\
D_x(v_i^k)=D_x(m_i)=D_x(L)=0 \text{ for } i=1,\dots,N;\, k=0,\dots,n \bigg\},
\endaligned
\tag 3.21
$$
where $D_x(\,.\,)$ denotes the directional differentiation in the direction of
$x$. Clearly, the complex linear subspace $\neutr$ of $\Bbb C^{(2\nu+1)N+1}$
is the proper complex analog of the neutral space of a genuine, real orbit
segment, see \S2.
\subheading{Remark} Since $\delta v_i^0$, $\delta m_i$, and $\delta L$
must be equal
to zero for a neutral vector $x\in\neutr$, we shall simply write
$x=(\delta\tilde q_1^0,\dots,\delta\tilde q_N^0)$ instead of indicating the
zero entries.
The proof of the following proposition is completely analogous to that one for
the real case, and therefore we omit it.
\subheading{Proposition 3.22} For every tangent vector $x\in\neutr$
($\omega\in\Omega\harmas$) and $1\le k\le n$ we have
$$
D_x\left(\tilde q_{i_k}^{k-1}-\tilde q_{j_k}^{k-1}\right)=\left(\alpha_k(x)-
\alpha_{k-1}(x)\right)\cdot \left(v_{i_k}^{k-1}-v_{j_k}^{k-1}\right),
\tag 3.23
$$
$$
D_x(t_k)=-\alpha_k(x),
\tag 3.24
$$
and
$$
D_x\left(\tau_k\right)=D_x(t_k-t_{k-1})=\alpha_{k-1}(x)-\alpha_{k}(x),
\tag 3.25
$$
where, by definition, $t_0=0=\alpha_0(x)$. The functions
$\alpha_k:\, \neutr\to\Bbb C$ ($k=1,\dots,n$) are linear functionals.
The name of the functional $\alpha_k$ is the {\it advance of the $k$-th
collision} $\sigma_k=\{i_k,j_k\}$, see also Section 2.
\medskip
This result markedly shows that, indeed, the vector space $\neutr$
describes the linearity of the $\harmas$-iterated billiard map.
\subheading{Remark 3.26} It is a matter of simple computation to convince
ourselves that all assertions of Section 2 of the present paper and
[Sim(1992)-II] pertaining to the neutral space $\neutr$ remain valid for the
complexified dynamics. Here we only point out the three most important
statements out of those results:
(1) The vector space $\neutr$ measures the ambiguity in recovering the orbit
segment $\omega$ purely from its velocity history
$\left\{v_i^k(\omega):\, i=1,\dots,N;\, k=0,\dots,n \right\}$ and the outer
geometric data $(\vec m,L)$. In other words,
this means that if, locally in the phase space $\Omega$ (in a small open
set), two phase points $\omega_1$ and $\omega_2$ have the same velocities
($v_i^k(\omega_1)=v_i^k(\omega_2)$, $i=1,\dots,N$; $k=0,\dots,n$) and
geometric parameters $(\vec m,L)$,
then the initial data of $\omega_1$ and $\omega_2$ can only differ by a
spatial translation by a vector from the space
$\Cal N(\omega_1)=\Cal N(\omega_2)$, and this statement is obviously
reversible.
(2) The reflection laws for the neutral vectors are exactly the same as for
velocities.
(3) The Connecting Path Formula (CPF, i. e. Lemma 2.9 of [Sim(1992)-II]
or Proposition 2.19 in [S-Sz(1999)]) is applicable, and it enables us to
compute the neutral space $\neutr$ via solving a homogeneous system of linear
equations (now over the field $\Bbb C$)
$$
\sum\Sb k=1\endSb\Sp n\endSp \alpha_k\Gamma_{ik}=0,\; i=1,\dots,n+P_{\Sigma}-N,
\tag 3.27
$$
where each equation in (3.27) is a $\nu$-dimensional complex vector equation,
the coefficients $\Gamma_{ik}=\Gamma_{ik}(\omega)\in\Bbb C^{\nu}$ are certain
linear combinations of relative velocities $v_i^k(\omega)-v_j^k(\omega)$
(the coefficients of those linear combinations are just fractional linear
expressions of the masses, see 2.17 and 2.18 in [S-Sz(1999)]) and, finally,
$P_{\Sigma}$ denotes the number of connected components of the collision graph
of $\Sigma$. For a bit more detailed explanation of (3.27), see Remark 3.41 below.
\medskip
\subheading{Remark 3.28}
We note here that the ordering of the moments of collisions $t(f_i)$
(which plays a significant role in the CPF) is no longer
meaningful over the unordered field $\Bbb C$. However, the use of the
corresponding inequalities in the CPF is purely technical/notational. Those
inequalities only serve to introduce the combinatorial ordering of the
collisions $f_i$ (which are $\sigma$'s) prescribed by the indexes of the symbolic
collision sequence $\Sigma=(\sigma_1,\sigma_2,\dots ,\sigma_n)$.
\medskip
\subheading{Remark 3.29} As a straightforward consequence of the equation
(3.6), we obtain that $v_i^{k-1}=v_i^k$ for $i=1,\dots,N$ (i. e. the $k$-th
reflection does not change the compound velocity) if and only if
$$
\left\langle v_{i_k}^{k-1}-v_{j_k}^{k-1};\, \tilde q_{i_k}^k-\tilde q_{j_k}^k-
L\cdot a_k\right\rangle=0,
$$
and this is just the case of a tangential collision. Easy calculation shows
that the mentioned tangentiality occurs if and only if the two roots
$\tau_k$ of (3.8) coincide.
\medskip
It is obvious that $\dimc\neutr$ is at least $\nu+1$ (as long as not all
velocities are the same, which we can assume, for
$v_1^k(\omega)=\dots=v_N^k(\omega)$ occurs on a set of codimension $\ge2$),
because the flow direction and the uniform spatial
translations are necessarily contained in $\neutr$.
\medskip
\subheading{Definition 3.30. (Definition of sufficiency or
``geometric hyperbolicity'')}
The orbit segment $\omega\in\Omega\harmas$ is said to be sufficient if and
only if
\newline
$\dimc\neutr=\nu+1$.
\medskip
Finally, the main result of [S-Sz(1999)] (Key Lemma 3.33 below) will use the
following
\subheading{Definition 3.31} We say that the triple $\harmas$ has Property (A)
if and only if the following assertion holds:
For every pair of indices $1\le k\nu+1 \right) \\
\Longrightarrow \; P_1(\vec x)=P_2(\vec x)=\dots=P_s(\vec x)=0.
\endaligned
\tag 3.37
$$
We note here that the plynomials in (3.37) are actually real polynomials,
see Proposition 3.41/c below. This note gains a particular importance when
-- in \S4 -- we switch from the complex case back to the real one.
\medskip
\subheading{Remark 3.38} The left-hand-side in the above implication
(i. e. the premise) precisely says that {\it some} branch of the multiple-valued
$\harmas$-dynamics with initial data $\vec x$ is not sufficient, see also
Remark 2 to Definition 3.18.
\subheading{Proof} This proof is a typical application of the Connecting
Path Formula (see \S2 here or Lemma 2.9 of [Sim(1992)-II])
and Proposition 3.4 of [Sim(1992)-II]. Namely, Proposition 3.4 of
[Sim(1992)-II] asserts that
$$
\dimc\neutr=\nu\cdot P_{\Sigma}+\dimc\{\alpha_1,\dots,\alpha_n\},
\tag 3.39
$$
where, as said before, $P_{\Sigma}$ denotes the number of connected components
of the collision graph of $\Sigma$, and $\{\alpha_1,\dots,\alpha_n\}$ is a
shorthand notation for the complex linear space of all possible $n$-tuples
$\left(\alpha_1(x),\dots,\alpha_n(x)\right)$ of advances, $x\in\neutr$.
\subheading{Remark 3.40} It is worth noting here that our present formula
(3.39) differs from its counterpart in Proposition 3.4 of [Sim(1992)-II]
by an additional term $\nu$. This is, however, due to the fact that in the
present approach we no longer have the reduction equation
$\sum_{i=1}^N m_i\delta\tilde q_i=0$.
It follows easily from (3.39) that the sufficiency is equivalent to
$\dimc\{\alpha_1,\dots,\alpha_n\}=1$, i. e. sufficiency means that all
advances are equal to the same functional.
Note that $\dimc\{\alpha_1,\dots,\alpha_n\}=1$ obviously implies
$P_{\Sigma}=1$.
\medskip
A simple, but important consequence of the Connecting Path Formula is that
the linear space $\{\alpha_1,\dots,\alpha_n\}$ is the solution set of a
homogeneous system of linear equations (3.27).
\subheading{Remark 3.41}
Note that, when applying Lemma 2.9 of [Sim(1992-II)], the left-hand-side of the
Connecting Path Formula has to be written as the relative velocity of the
colliding balls multiplied by the advance of that collision. It follows
immediately from the exposition of [Sim(1992)-II] that the equations of
this sort (arising from all CPF's) are the {\it only} constraints on the
advance functionals. The reason why this is true is that the fulfillment
of all CPF's precisely means that the relative displacement (variation of
position) of every pair of particles right before their new collision is
parallel to the relative incoming velocity of these particles (see also
Proposition 3.22), and this fact guarantees that the variation of the newly
formed relative outgoing velocity will also be zero, just as required by
(3.21).
Also note that the number $n+P_{\Sigma}-N$ of equations
in (3.27) is equal to the number of all collisions
$\sigma_k=\{i_k,j_k\}$ for which $i_k$ and $j_k$ are in the same connected
component of the collision graph of
$\{\sigma_1,\sigma_2,\dots ,\sigma_{k-1}\}$.
\medskip
The determinant $D(M)$ of every minor $M$ of the
coefficient matrix of (3.27) is a homogeneous velocity polynomial and,
therefore, $D(M)$ is a holomorphic function on $\Omega\harmas$.
Thus, there are finitely many velocity polynomials (actually, determinants)
$R_1(\omega),\dots,R_s(\omega)$ such that their simultaneous vanishing follows
from the non-sufficiency of the orbit segment $\omega$. By using the
equations (3.4)-(3.8) (which define the complex dynamics recursively),
each of these velocity polynomials $R_i(\omega)$ can be written as an algebraic
function
$f_i\left(\vec x(\omega)\right)=f_i(\vec x)$ of the initial data $\vec x$,
and these algebraic functions $f_i(\vec x)$ only contain (finitely many) field
operations and square roots. By using the canonical method of successive
elimination of the square roots from the equation $f_i(\vec x)=0$, one obtains
a complex polynomial $P_i(\vec x)$ such that
$$
P_i(\vec x)=0\Longleftrightarrow f_i(\vec x)=0.
$$
We emphasize here that
\roster
\item we understand the relation $f_i(\vec x)=0$ in such
a way that $f_i(\vec x)$ becomes zero on {\it some} branch of the square
root function when evaluating the multiple-valued algebraic function
$f_i(\vec x)$;
\item the equivalence is claimed to only hold on $D$, cf. the formulation
of Lemma 3.36.
\endroster
The process of eliminating the square roots from
an equation $f(\vec x)=0$ (of the above type), however, requires a bit of
clarification. The algebraic function $f$ has a natural representing element
$\hat f$ in the field $\Bbb K_n$, see Definition 3.11. Consider the normal
hull $\overline{\Bbb K}_n:\Bbb K_0$ of the field extension
$\Bbb K_n:\Bbb K_0$. In other words, let $\overline{\Bbb K}_n$
be the splitting field (over $\Bbb K_0$) of the minimal polynomial
$\phi(x)=x^k+\sum_{i=0}^{k-1}a_ix^i$ of a generator $\alpha\in\Bbb K_n$
of the extension $\Bbb K_n$ over the field $\Bbb K_0=\Bbb C(\vec x)$.
(We note here that, since the extension $\Bbb K_n:\Bbb K_0$ is a simple
algebraic extension, there exists such an element $\alpha\in\Bbb K_n$ with
$\Bbb K_n=\Bbb K_0(\alpha)$, i. e. the field $\Bbb K_n$ can be obtained by
a simple adjunction of the element $\alpha$ to the field $\Bbb K_0$.)
The field $\overline{\Bbb K}_n$ contains all
conjugates $\hat f^{(1)},\,\dots,\,\hat f^{(k)}$ of the element
$\hat f=\hat f^{(1)}$ over the field $\Bbb K_0$. Let
$$
N(\hat f)=\prod_{i=1}^k \hat f^{(i)}=a_0\in\Bbb K_0=\Bbb C(\vec x)
\tag 3.41/a
$$
be the norm of the element $\hat f$ over $\Bbb K$. (For the elementary
concepts and facts from Galois theory, the reader is kindly referred to
the books [St(1973)], [VDW(1970)].) The element $N(\hat f)$ has the form
$$
N(\hat f)=\frac{P(\vec x)}{Q(\vec x)}\in\Bbb K_0
\tag 3.41/b
$$
with uniquely determined (up to scalar multipliers) relatively prime complex
polynomials $P(\vec x)$ and $Q(\vec x)$.
Actually, as it directly follows from Proposition 3.3 and from the method
of eliminating the square roots (presented above), in each denominator
$Q_l(\vec x)$ ($1\le l\le s$) associated to the algebraic function
$f_l(\vec x)$ there are only factors of the type
$m_i+m_j$ or $\Vert v_i^k-v_j^k \Vert^2$, $1\le i0$ for which
$N\notin\sigma_i$.)
\subheading{Remark} It is worth noting that, as it follows easily from the
conditions, $v_N^p(\omega)=v_N^{q-1}(\omega)$.
\medskip
\subheading{Proof} First of all, we observe that the motion of the $N$-th
ball with zero mass has absolutely no effect on the evolution of the
$\{1,\dots,N-1\}$-part of the trajectory segment, and this statement is also
valid for the time evolution of the $\{1,\dots,N-1\}$-part
$\left\{\delta\tilde q_i^k|\, i=1,\dots,N-1\right\}$ of a neutral vector
$\left(\delta\tilde q_1^0,\dots,\delta\tilde q_N^0\right)\in\neutr$,
see also the equations (3.4), (3.7)--(3.8) and (3.10).
\medskip
Consider now an arbitrary neutral vector
$\delta Q=\left(\delta\tilde q_1^0,\dots,\delta\tilde q_N^0\right)\in\neutr$.
According to (1) and the above principle, we can modify the neutral vector
$\delta Q$ by a scalar multiple of the flow direction and by a uniform
spatial displacement in such a way that $\delta\tilde q_i^0=0$ for
$i=1,\dots,N-1$, and then $\delta\tilde q_i^k=0$ remains true for the whole
orbit segment, $i=1,\dots,N-1$, $k=0,\dots,n$. The neutrality of $\delta Q$
with respect to the entire orbit segment means, however, that
$$
\delta\tilde q_N^p=\alpha_p\cdot\left(v_N^p(\omega)-v_{i_p}^p(\omega)\right)
\text{ and }\delta\tilde q_N^{q-1}=\alpha_q\cdot\left(v_N^{q-1}(\omega)-
v_{i_q}^{q-1}(\omega)\right).
\tag 3.45
$$
According to our hypothesis (2), the consequence of (3.45) and of the obvious
equation $\delta\tilde q_N^p=\delta\tilde q_N^{q-1}$ is that
$\alpha_p=\alpha_q=0$, and thus $\delta Q=0$. Hence the lemma follows. \qed
\medskip
The crucial part of the inductive proof of Key Lemma 3.33, i. e. the
substitution $m_N=0$, requires some preparatory thoughts and lemmas.
Assume that a
combinatorial-algebraic scheme $\harmas$ is given for the $N$-ball system
$\{1,2,\dots,N\}$ so that Property (A) holds (see Definition 3.31).
In the proof of Key Lemma 3.33 we want to use some
combinatorial-algebraic ($N-1$)-schemes $\harmasv$ that
(i) govern the time evolution of the $\{1,2,\dots,N-1\}$-part of {\it some}
$\harmas$-orbit segments with an infinitely light $N$-th ball, i. e.
$m_N=0$ and
(ii) enjoy Property (A).
In the construction of such $(N-1)$-schemes $\harmasv$, the so called
{\it derived schemes}, we want to follow the guiding principles below, which
also serve as the definition of the {\it derived schemes:}
\medskip
\subheading{Definition 3.46} We say that the $(N-1)$-scheme $\harmasv$
is a scheme derived from $\harmas$ by putting $m_N=0$ if it is obtained
as follows:
(1) First of all, we discard all symbols $\sigma_j$, $a_j$, and $\tau_j$
from $\harmas$ that correspond to a $\sigma_j$ containing the label $N$.
This also means that we retain the other symbols $\sigma_j$, $a_j$ without
change: they only get re-indexed, due to the dropping of the other symbols;
\medskip
(2) As far as the selection of $\vec\tau'$ is concerned, we want to retain
{\it all algebraic relations} among the variables that are encoded in the
original scheme $\harmas$, i. e. all algebraic relations that follow from
the scheme $\harmas$ and from $m_N=0$;
\medskip
(3) We want the derived scheme $\harmasv$ to enjoy Property (A), see 3.31.
\medskip
It is straightforward that, following just the instructions in (1)-(2)
above, one can easily construct such schemes $\harmas$ which, perhaps,
do not have Property (A).
\bigskip
\subheading{Remark 3.47} Part (2) of the above definition means the
following: Initially, the algebra of $\harmas$ consists
\roster
\item of the variables
$$
\left\{\left(\tilde q_i^k\right)_j,\, (v_i^k)_j,\, m_i,\, L,\,
\tau_l \big| \, i=1,\dots,N;\, k=0,\dots,n;\, j=1,\dots,\nu;\,
l=1,\dots,n \right\};
$$
\item of the equations (3.4-8), and finally
\item of the choices of the ``signs of the square roots''
in the solutions of the quadratic equations (3.8) whenever the equations
are reducible (cf. Remark 3.13).
\endroster
To obtain the algebra of the derived scheme $\harmasv$, we put first
$m_N=0$ and then $\tau'_l=\sum_{j=k+1}^l \tau_j$
whenever for some $1\le k0$ for $k=1,\dots,n$;
(3) out of the two real roots of (3.8) the root $\tau_k$ is always selected
as the smaller one, $k=1,\dots,n$.
\medskip
It is clear that either $\Omega_{\Bbb R}=\Omega_{\Bbb R}\harmas$ is a
$\left((2\nu+1)N+1\right)$-dimensional, real analytic submanifold of
$\Omega=\Omega\harmas$, or $\Omega_{\Bbb R}=\emptyset$. Of course, we will
never investigate the case $\Omega_{\Bbb R}=\emptyset$.
Consider the corresponding polynomials $P_1(\vec x),\dots,P_s(\vec x)$ of
(3.37) describing the non-sufficiency of the complex orbit segments
$\omega\in\Omega\harmas$ along the lines of Lemma 3.36 in terms of the kinetic data
$\vec x=\vec x(\omega)$. According to Proposition 3.41/c, these polynomials
$P_i(\vec x)$ admit real coefficients. By Key Lemma 4.10, the greatest common
divisor of $P_1(\vec x),\dots,P_s(\vec x)$ is $1$, hence the common zero set
$$
\left\{\vec x\in\Bbb R^{(2\nu+1)N+1}\big|\; P_1(\vec x)=P_2(\vec x)=\dots
=P_s(\vec x)=0\right\}
$$
of these polynomials does not contain any smooth real submanifold of
(real) dimension $(2\nu+1)N$. In this way we obtained
\medskip
\subheading{Proposition 5.1} Use all the notions, notations and assumptions
from above. There exists no smooth, real submanifold $\Cal M$ of
$\Omega_{\Bbb R}$ with
$\text{dim}_{\Bbb R}\Cal M=\text{dim}_{\Bbb R}\Omega_{\Bbb R}-1
\left(=(2\nu+1)N\right)$ and with the property that all orbit segments
$\omega\in\Cal M$ are non-sufficient. (For the concept of non-sufficiency,
please see \S2.) \qed
\bigskip
\subheading{The ``Fubini-type'' Argument}
\medskip
Our dynamics $\Omega\harmas$ has the obvious feature that the variables
$m_i=m_i(\omega)$ ($i=1,\dots,N$) and $L(\omega)$ (the so called outer
geometric parameters) remain unchanged during the time-evolution. Quite
naturally, we do not need Proposition 5.1 directly but, rather, we need to
use its analog for (almost) every fixed $(N+1)$-tuple
$(m_1,\dots,m_N;L)\in\Bbb R^{N+1}$. This will be easily achieved by a
classical ``Fubini-type'' product argument. The result is
\medskip
\subheading{Proposition 5.2} Use all the notions, notations and assumptions
from above. Denote by
$$
NS=NS\harmas=\left\{\omega\in\Omega_{\Bbb R}\harmas\big|\;
\text{dim}_{\Bbb C}\Cal N(\omega)>\nu+1\right\}
$$
the set of all non-sufficient, real orbit segments
$\omega\in\Omega_{\Bbb R}=\Omega_{\Bbb R}\harmas$. (For the definition of the
complex neutral space $\Cal N(\omega)$, please see (3.21).) Finally, we use
the notation
$$
\Omega_{\Bbb R}(\vec m,\,L)=\left\{\omega\in\Omega_{\Bbb R}\big|\;
\vec m(\omega)=\vec m, \text{ and } L(\omega)=L\right\}
$$
for any given $(N+1)$-tuple
$(\vec m,\,L)=(m_1,\dots,m_N,\,L)\in\Bbb R^{N+1}$. We claim that for almost
every $(\vec m,\,L)\in\Bbb R^{N+1}$ (according to the Lebesgue measure) the
intersection $NS\cap\Omega_{\Bbb R}(\vec m,\,L)$ has at least $2$
codimensions in $\Omega_{\Bbb R}(\vec m,\,L)$.
\medskip
\subheading{Remark 5.3} As it is always the case with such algebraic systems,
the exceptional zero-measure set of the parameters $(\vec m,\,L)$ turns out
to be a countable union of smooth, proper submanifolds of $\Bbb R^{N+1}$.
\medskip
\subheading{Proof of 5.2} It is clear that the statement of the proposition
is a local one, therefore it is enough to prove that for any small, open
subset $U_0\subset\Omega_{\Bbb R}$ of
$\Omega_{\Bbb R}=\Omega_{\Bbb R}\harmas$ the set
$$
\left\{(\vec m,\,L)\in\Bbb R^{N+1}\big|\; \text{dim}_{\Bbb R}\left(
NS\cap\Omega_{\Bbb R}(\vec m,\,L)\cap U_0\right)\ge 2\nu N-1\right\}
$$
of the ``bad points'' $(\vec m,\,L)$ has zero Lebesgue measure. The points
$\omega\in U_0$ can be identified locally (in $U_0$) with the vector
$\vec x=\vec x(\omega)\in D_{\Bbb R}=D\harmas\cap\Omega_{\Bbb R}$ of their
coordinates. After this identification the small open set
$U_0\subset\Omega_{\Bbb R}$ naturally becomes an open subset
$U_0\subset D_{\Bbb R}$. Furthermore, we split the points $\vec x\in U_0$
as $\vec x=\left((\vec m,\,L),\,\vec y\right)$, where $\vec y$ contains all
variables other than $m_1,\,\dots,\,m_N,\,L$. In this way we may assume that
$U_0$ has a product structure $U_0=B_0\times B_1$ of two small open balls,
so that $B_0\subset\Bbb R^{N+1}$, while the open ball
$B_1\subset\Bbb R^{2\nu N}$ contains the $\vec y$-parts of the points
$\vec x=\left((\vec m,\,L),\,\vec y\right)\in U_0$.
Assume that the statement of the proposition is false. Then there exists
a small open set $U_0=B_0\times B_1\subset\Bbb R^{N+1}\times\Bbb R^{2\nu N}$
(with the above splitting) and there is a positive number $\epsilon_0$ such
that the set
$$
\aligned
A_0=&\big\{(\vec m,\,L)\in B_0\big|\; \left((\vec m,\,L)\times B_1\right)
\cap NS\text{ contains a} \\
&(2\nu N-1)\text{-dimensional, smooth, real submanifold with inner radius }
>\epsilon_0\big\}
\endaligned
$$
has a positive Lebesgue measure in $B_0$. Then one can find an orthogonal
projection $P:\; \Bbb R^{2\nu N}\to H$ onto a hyperplane $H$ of
$\Bbb R^{2\nu N}$ such that, by taking
$\Pi(\vec x)=\Pi\left((\vec m,\,L),\,\vec y\right)=P(\vec y)$
($\Pi:\; \Bbb R^{(2\nu+1)N+1}\to H$), the set
$$
\aligned
A_1=&\big\{(\vec m,\,L)\in B_0\big|\; \Pi\left[\left((\vec m,\,L)
\times B_1\right)\cap NS\right]\text{ contains} \\
&\text{an open ball of radius }>\epsilon_0/2\text{ in } H\big\}
\endaligned
$$
has positive Lebesgue measure in $B_0$. By the Fubini Theorem the set
$$
\tilde\Pi\left[NS\cap(B_0\times B_1)\right]
$$
has positive Lebesgue measure in $B_0\times H$, where
$$
\tilde\Pi(\vec x)=\tilde\Pi\left((\vec m,\,L),\,\vec y\right)=
\left((\vec m,\,L),\,P(\vec y)\right)\in B_0\times H
$$
for $\vec x\in B_0\times B_1$. However,
$\text{dim}_{\Bbb R}(B_0\times H)=(2\nu+1)N$, and
$\text{dim}_{\Bbb R}\left(NS\cap\Omega_{\Bbb R}\right)\le(2\nu+1)N-1$
(according to Proposition 5.1). Thus, we obtained that the real algebraic set
$$
\tilde\Pi\left(NS\cap(B_0\times B_1)\right)\subset B_0\times H
$$
has dimension strictly less than
$\text{dim}_{\Bbb R}(B_0\times H)=(2\nu+1)N$, yet it has positive Lebesgue
measure in the space $B_0\times H$. The obtained contradiction finishes the
proof of Proposition 5.2. \qed
\bigskip
\heading
Finishing the Proof of the Theorem
\endheading
\bigskip
We will carry out an induction with respect to the number of balls $N$
($\ge2$). For $N=2$ the system is well known to be a strictly dispersive
billiard flow (after the obvious reductions $m_1v_1+m_2v_2=0$,
$m_1||v_1||^2+m_2||v_2||^2=1$ ($m_1,\,m_2>0$), and after the factorization
with respect to the uniform spatial translations, as usual) and, as such,
it is proved to be ergodic by Sinai in [Sin(1970)], see also the paper
[S-W(1989)] about the case of different masses.
Assume now that $N\ge3$, $\nu\ge3$, and the theorem has been successfully
proven for all smaller numbers of balls $N'0$, $L>0$) in such a way that,
besides the always assumed properties (2.1.1)--(2.1.2),
\medskip
(*) the vector $(\vec m,\,L)$ of geometric parameters is such that for any
proper subsystem $1\le i_1