\input amstex.tex
\documentstyle{amsppt}
\magnification=\magstep 1
\TagsAsMath
\define\sdb{semi-dispersing billiard}
\define\sdbs{semi-dispersing billiards}
\define\bil{billiard}
\define\sinb{\hbox{[Sin(1970)]\;}}
\define\sci{(\Sigma,C_i)}
\def\bt{{\Bbb T}}
\def\br{{\Bbb R}}
\def\ba{{\Bbb A}}
\def\bz{{\Bbb Z}}
\def\boq{{\bold {Q}}}
\def\bn{{\Bbb N}}
\define\coK{K_{(\bar q_i,I_i,E_i)}}
\define\coKt{K_{(\bar q_i +tI_i,I_i,E_i)}}
\def\ks{\Cal S}
\def\kr{\Cal R}
\define\sr{\ks\kr^+}
\def\g{\gamma}
\define\spix{S^{(0,\infty)}x}
\def\bp{\Bbb P}
\def\kt{\Cal T}
\def\kv{\Cal V}
\def\ke{\Cal E}
\define\clc{C\ell_C (C_{ed}(t_0,P))}
\define\ctp{C_{ed}(t_0,P)}
\define\xu{U(x_0)}
\define\yu{U(y_0)}
\define\fie{\uph _{I,E}\,}
\define\pik{\pi_{2,3,4}\,}
\define\kssa{\hbox{[K-S-Sz (1989)]\;}}
\define\kssb{\hbox{[K-S-Sz (1990)]\;}}
\define\kssc{\hbox{[K-S-Sz (1991)]\;}}
\define\kssd{\hbox{[K-S-Sz (1992)]\;}}
\define\sima{\hbox{[Sim(1992)-I]\;}}
\define\simb{\hbox{[Sim(1992)-II]\;}}
\define\szaa{\hbox{[Sz (1995)]\;}}
\define\szab{\hbox{[Sz (1994)]\;}}
\define\ssz{\hbox{[S-Sz (1994)]\;}}
\define\sc{\hbox{[S-Ch (1987)]\;}}
\define\bus{\hbox{[B-S (1973)]\;}}
\define\blps{\hbox{[BLPS (1992)]\;}}
\define\dist{\text{dist\;}}
\define\nt{\text{int\;}}
\define\vtm{v^{t^-}}
\define\vTm{v^{T^-}}
\define\vtp{v^{t^+}}
\define\vTp{v^{T^+}}
\define\wtm{w^{t^-}}
\define\wtp{w^{t^+}}
\define\wTm{w^{T^-}}
\define\wTp{w^{T^+}}
\def \t {\text {\bf T}}
\def \r {\Bbb R}
\def \q {\text {\bf Q}}
\def \z {\text {\bf Z}}
\def \T {\text {\bold {T}}}
\def \tf {\bold {T}^4}
\def \qf {(q_1,q_2,q_3,q_4)}
\define\sinf{S^{(-\infty,\infty)}x_0\,}
\define\xim{$x_0\in M_d$\,}
\define\mpp{M_{P_1,P_2}\,}
\define\fcf{F_{-}\cap F_{+}\,}
\define\sncx{$S^{(-\infty ,0)}_{C_i(P_1)}x_0$\,}
\define\spcx{$S^{(0,\infty )}_{C_i(P_2)}x_0$\,}
\define\sncy{$S^{(-\infty ,0)}_{C_i(P_1)} y$\,}
\define\spcy{$S^{(0,\infty )}_{C_i(P_2)} y$\,}
\define\steb{\left\{ S^t_{1,2}\right\}\,}
\define\sthb{\left\{ S^t_{3,4}\right\}\,}
\define\stnb{\left\{ S^t_{2,3,4}\right\}\,}
\define\ste{S^t_{1,2}\,}
\define\sth{S^t_{3,4}\,}
\define\stn{S^t_{2,3,4}\,}
\define\pikv{\pi^V_{2,3,4}\,}
\define\sni{S^{(-\infty,0)}\,}
\define\snis{S^{(-\infty,0)}_*\,}
\define\snin{S^{(-\infty,0)}_{2,3,4}\,}
\define\spi{S^{(0,\infty)}\,}
\define\spis{S^{(0,\infty)}_*\,}
\define\spie{S^{(0,\infty)}_{1,2}\,}
\define\spih{S^{(0,\infty)}_{3,4}\,}
\define\spin{S^{(0,\infty)}_{1,3,4}\,}
\define\vk{(v_1,v_2,v_3,v_4)\,}
\define\qvi{(q_1,v_1,I-v_1)\,}
\define\qvk{(q_1,q_2,v_1,v_2)\,}
\define\qvio{(q_1^0,v_1,I-v_1)\,}
\define\qviz{(q_1^0, v_1^0, I-v_1^0)\,}
\define\bbb{\bold {B}}
\define\aaa{\bold {A}}
\define\Q{\bold {Q}}
\define\qm{\partial \bold {Q}^-}
\define\qp{\partial \bold {Q}^+}
\define\qi{\partial \bold {Q}_i}
\define\mi{\partial {M}_i}
\define\mpl{\partial {M}^+}
\define\mm{\partial {M}^-}
\define\suf{sufficient\ \ }
\define\erg{ergodicity\ \ }
\define\traj{S^{[0,T]}x_0}
\define\flow{\left(\bold{M},\{S^t\},\mu\right)}
\define\proj{\Bbb P^{\nu-1}(\Bbb R)}
\define\ter{\Bbb S^{d-1}\times\left[\Bbb P^{\nu-1}(\Bbb R)\right]^m}
\define\sphere{\Bbb S^{d-1}}
\define\pont{(V;h_1,h_2,\dots ,h_m)}
\define\projm{\left[\Bbb P^{\nu-1}(\Bbb R)\right]^m}
\define\pontg{(V_0;g_1,g_2,\dots ,g_m)}
\define\szorzat{\prod\Sb i=1\endSb\Sp m\endSp \Cal P_i}
\define\szorzatk{\prod\Sb l=1\endSb\Sp k\endSp \Cal P_{i_l}}
\define\ball{\overline{B}_{\varepsilon_0}(x_0)}
\define\qv{(Q,V^+)}
\document
{\catcode`\@=11\gdef\logo@{}}
%\advance\baselineskip 8pt
\noindent
February 21, 1998
\bigskip \bigskip
\centerline{\bf Cylindric Billiards and Transitive Lie Group Actions}
\bigskip \bigskip \bigskip
\centerline{{\bf N\'andor Sim\'anyi}
\footnote{Research supported by the Hungarian National Foundation for
Scientific Research, grants OTKA-7275 and OTKA-16425.}}
%\footnote{On leave from the Bolyai Institute of the University of
%Szeged, Aradi V\'ertanuk tere 1, Szeged, H-6720 Hungary. E-mail:
%simanyi\@math.u-szeged.hu}}
\centerline{Bolyai Institute, University of Szeged,}
\centerline{6720 Szeged, Aradi V\'ertanuk tere 1, Hungary.}
\centerline{E-mail: simanyi\@math.u-szeged.hu}
\hbox{}
\centerline {\bf Domokos Sz\'asz$^1$}
\centerline{Mathematical Institute of the Hungarian Academy of Sciences}
\centerline{H-1364, Budapest, P. O. B. 127, Hungary}
\centerline{E-mail: szasz\@math-inst.hu}
\bigskip \bigskip
\heading
1. Introduction.
\endheading
\bigskip \bigskip
Semi-dispersive billiards is a class of billiards evidently
without any ellipticity, actually with more or less manifest
hyperbolicity; their study was initiated by Chernov and Sinai in 1987
[S-Ch(1987)]. {\it
Cylindric billiards}, a much interesting subfamily of semi-dispersive
billiards were introduced in 1992 by the second author,
[Sz(1993)]. Cylindric billiards are interesting for,
on one hand, they contain {\it hard ball systems},
a fundamental model from
the aspects of statistical physics and a much beautiful one, we believe,
from the point of view of mathematics; and
on the other hand, this is apparently the widest subclass of
semi-dispersive billiards where the search for {\it transparent necessary and
sufficient conditions of ergodicity} is promising.
In words, cylindric billiards are toric billiards where the scatterers
are cylinders. In our discussion, the bases of the cylinders will be
assumed to be stricly convex, a property ensuring that the scatterers be
convex, and thus the arising billiard be semi-dispersive. Because of
the simplicity of our model, let us immediately start with a formal
definition.
The configuration space of a cylindric billiard is
$\bold Q=\Bbb T^d\setminus\left(C_1\cup\dots\cup C_k\right)$, where
$\Bbb T^d=\Bbb R^d/\Bbb Z^d$ ($d\ge 2$) is the unit torus. Here the
cylindric scatterer $C_i$ ($i=1,\dots ,k$) is defined as follows:
Let $A_i\subset\Bbb R^d$ be a so called {\it lattice subspace} of the
Euclidean space $\Bbb R^d$, which means that the discrete intersection
$A_i\cap \Bbb Z^d$ has rank $\dim A_i$. In this case the factor
$A_i/(A_i\cap\Bbb Z^d)$ naturally defines a subtorus of $\Bbb T^d$,
which will be taken as the generator of the cylinder
$C_i\subset\Bbb T^d$. Denote by $L_i=A_i^\perp$ the orthocomplement
of $A_i$. Under the above conditions the subspace $L_i$ must also be a
lattice subspace. Throughout this article we will always assume that
$\dim L_i\ge 2$. Let, moreover, $B_i\subset L_i$ be a convex,
compact domain with a $C^2$-smooth boundary $\partial B_i$ containing
the origin as an interior point. We will assume that $B_i$ is strictly
convex in the sense that the second fundamental form of its boundary
$\partial B_i$ is everywhere positive definite. Furthermore, in order
to avoid unnecessary complications, we assume that the convex domain
$B_i$ does not contain any pair of points congruent modulo $\Bbb Z^d$.
The domain $B_i$ will be taken as the {\it base} of the cylinder $C_i$.
Finally, suppose that a translation vector $t_i\in\Bbb R^d$ is given,
playing an essential role in positioning the cylinder $C_i$ in the
ambient torus $\Bbb T^d$. Set
$$
C_i=\left\{a+l+t_i\, |\; a\in A_i,\; l\in L_i,\; \right\}\big/\Bbb Z^d.
$$
In order to avoid further unnecessary complications, we also assume that
the interior of the configuration space
$\bold Q=\Bbb T^d\setminus\left(C_1\cup\dots\cup C_k\right)$ is connected.
The phase space $\bold M$ of our billiard will be
the unit tangent bundle of $\bold Q$, i. e.
$\bold M=\bold Q\times\Bbb S^{d-1}$.
(Here, as usual, $\Bbb S^{d-1}$ is the $d-1$-dimensional unit sphere.)
The dynamical system $(\bold M, S^{\Bbb R},\mu)$,
where $S^{\Bbb R}$ is the dynamics defined
by uniform motion inside the domain and specular reflections at its
boundary (the scatterers!) and $\mu$ is the Liouville measure, is
a {\it cylindric billiard} we want to study. (As to notions and notations
in connection with semi-dispersive billiards the reader is recommended
to consult the work [K-S-Sz(1990)].)
In 1994, the first general result: necessary and sufficient conditions
of ergodicity, was obtained in [Sz(1994)] for the class of the
so-called {\it orthogonal} cylindric billiards, where the generator of
each cylinder is parallel to some coordinate subspace of the
orthogonal system of coordinates in which the underlying torus is given.
Then, in [S-Sz(1994)], sufficient conditions were found for the
ergodicity, in fact, for a hyperbolic one, of a
billiard with two non-orthogonal cylindric scatterers. The method of the
proof already revealed the intimate role which the {\it transitivity on
$\Bbb S^{d-1}$ of
the action of a Lie-subgroup $\Cal G$ of the orthogonal group $O(d)$}
would play in ensuring the hyperbolicity, and consequently
the ergodicity, too, of the cylindric billiard in question. To be more
precise,
for any cylinder $C_i$ ($i=1,\dots ,k$) consider the group $\Cal G_i$
consisting of all orthogonal transformations $U$ of $\Bbb R^d$ that
leave the points of the subspace $A_i$ fixed.
Then we consider the subgroup $\Cal G$
of $O(d)$ algebraically generated by all such groups $\Cal G_i: 1 \le
i \le k$, which is certainly an embedded Lie subgroup of $O(d)$.
Our aim in this work is to formulate precise statements about
the aforementioned transitivity of the action of $\Cal G$.
Having collected some necessary notions in section 2, the general
results are formulated in section 3. According to our Conjecture 1,
transitivity of $\Cal G$ is equivalent to the hyperbolic ergodicity of the
flow.
\medskip
\proclaim{Conjecture 1} For every cylindric billiard flow $\flow$
the transitivity of the action of $\Cal G$ on the velocity sphere
$\Bbb S^{d-1}$ is equivalent to the hyperbolic ergodicity (or, equivalently,
to the hyperbolicity and Bernoulli property) of the billiard map.
\endproclaim
\medskip
The main result of section 3 is
\proclaim{Theorem 3.6} The irreducibility of the $\Cal G$-action on $\Bbb R^d$
implies the transitivity on $\Bbb S^{d-1}$.
\endproclaim
This theorem has the following important corollary:
\proclaim{Corollary A} The transitivity of the $\Cal G$-action on
$\Bbb S^{d-1}$ is a necessary condition for the ergodicity of the
cylindric billiard.
\endproclaim
Though this corollary provides the necessity of the transitivity
property, it seems to be extremely hard to establish
the sufficiency of this condition. Indeed, for the
``simple'' subclass of cylindric billiards: hard ball systems, this is
not known, in general. In section 4 of the present work, we will
be able to show that for hard ball systems, transitivity of $\Cal G$
holds for any number $N$ of particles, in any dimension $\nu$, and for
arbitrary masses $m_1, \dots, m_N$ and radii $r$ of the particles.
As a contrast, we note that in our recent, quite involved paper [S-Sz(1998)],
we were only able
to prove hyperbolicity --- and still not ergodicity --- for
masses $m_1, \dots, m_N$ and radii $r$ of the particles not belonging to
a countable union of exceptional analytic submanifolds of parameters.
In the light of this last result, the merit of the transitivity statement
of section 4 mentioned above is that it holds for arbitrary geometric
parameters of the system --- without any exceptional set.
Motivated by the importance of the concept of transitivity, in section
3, we also introduce related notions: the tightness and the orthogonal
non-splitting property of $\Cal G$, the first of them being stronger than,
while the second one equivalent to transitivity.
Furthermore, Proposition 3.18 of Section 3 gives
a new, we think beatiful and quite surprising characterization of sufficiency.
\bigskip \bigskip
\heading
2. Prerequisites
\endheading
\bigskip \bigskip
As to the basic notions about semi-dispersive billiards we refer to the
paper [K-S-Sz (1990)]. For convenience and brevity, we will
throughout use the concepts and notations of Sections 2 and 3 of
that paper. Here we only summarize some further notions from
\kssc, \kssd, and [Sim(1992)]. These are either new or their
exposition is simpler than that given in the original work.
An often used abbreviation is the shorthand $S^{[a,b]}x$ for the
trajectory segment $\{S^tx: \, a\le t\le b\}$. The natural projections
from $\bold{M}$ onto its factor spaces are denoted, as usual, by
$\pi: \bold{M}\rightarrow \boq$ and
$p:\ \bold{M}\rightarrow\ks^{d-1}$ or, sometimes, we simply write
$\pi(x)=Q(x)=Q$ and $p(x)=V(x)=V$ for $x=(Q,V)\in\bold{M}$. Any
$t \in[a,b]$ with $S^tx \in \partial \bold M$ is called {\it a collision
moment or collision time}. Denote
$\partial\bold Q=\cup_{i=1}^k \partial\bold C_i$, where $\partial \bold C_i$
are the smooth
components of the boundary. Since we want to consider typical
situations, we will always assume that {\it at every point
$q\in\partial\bold Q$ of the boundary of the configuration space $\bold Q$
the spherical angle subtended by the compact domain $\bold Q$ is strictly
positive.}
As 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
[K-S-Sz(1990)]). An important consequence of Theorem 5.3 of [V(1979)] is
that --- for semi-dispersive billiards we are considering --- there are
{\it no trajectories
at all with a finite accumulation point of collision moments}.
As a result, for an arbitrary non-singular orbit segment $S^{[a,b]}x$ of
the cylindric billiard flow, there is a uniquely defined sequence
$a\le\tau_1<\tau_2<\dots<\tau_m\le b$ ($m\ge 0$) of
collision times and a uniquely defined sequence
$\sigma_1,\sigma_2,\dots ,\sigma_m$ ($1\le\sigma_i\le k$) of labels of
cylinders with the properties that
(i) for every $t\in[a,b]$,\, $S^tx\in\partial\bold M$ if and only if
$t=\tau_i$ with some $i=1,\dots ,m$,
\noindent and
(ii) $\pi\left(S^{\tau_i}x\right)\in\partial\bold C_{\sigma_i}$,\,
$i=1,\dots ,m$.
The sequence
$\Sigma:=\Sigma(S^{[a,b]}x):= (\sigma_1, \sigma_2, \dots, \sigma_m)$ is called
the {\it symbolic collision sequence} of the trajectory segment $S^{[a,b]}x$.
As well known, billiards are dynamical systems with {\it singularities}.
A collision at a
point $x\in\partial\bold M$ such that in $\pi(x)$, at least two smooth
pieces of $\partial{\text {\q}}$ meet,
is called a {\it multiple} collision.
A collision is called {\it tangential} if $x\in\partial\bold M$ and
$p(x)\in \Cal {T}_{\pi(x)}\partial{\bold {Q}}$, i. e.
$p(x)$ is tangential to $\partial \bold Q$ at the point of reflection.
We shall use the collection $\sr$ of all singular reflections:
\proclaim{Definition 2.1} The set $\sr$ is the collection of all phase points
$x\in\partial\bold M$ for which the reflection, occurring at $x$, is singular
(tangential or multiple) and, in the case of a multiple collision, $x$ is
supplied with the {\it outgoing} velocity $V^+$.
\endproclaim
It is not hard to see that $\sr$ is a compact cell-complex in $\bold M$ and
$\dim(\sr)=\dim\bold M-2=2d-3$.
\bigskip
\heading
Neutral Subspaces and Sufficiency
\endheading
\medskip
Consider a {\it non--singular} trajectory segment $S^{[a,b]}x$.
Suppose that $a$ and $b$ are {\it not moments of collision}.
Before defining the neutral linear space of this trajectory segment, we note
that the tangent space of the configuration space $\boq$ at
interior points can be identified with the common linear space
$\Bbb R^d$.
\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{s. t.} \; \forall \alpha \in (-\delta,\delta) \\
& p\left(S^a\left(Q(x)+\alpha W,V(x)\right)\right)=p(S^ax) \, \& \\
& p\left(S^b\left(Q(x)+\alpha W,V(x)\right)\right)=p(S^bx) \big \}.
\endaligned
$$
\endproclaim
It is known (see (3) in Section 3 of [S-Ch (1987)]) that
$\Cal N_0(S^{[a,b]}x)$ is a linear subspace of $\Bbb R^d$ indeed, and
$V(x)\in \Cal N_0(S^{[a,b]}x)$. The neutral space $\Cal N_t(S^{[a,b]}x)$
of the segment $S^{[a,b]}x$ at time $t\in [a,b]$ is defined as follows:
$$
\Cal N_t(S^{[a,b]}x)=\Cal N_0\left(S^{[a-t,b-t]}(S^tx)\right).
\tag 2.2
$$
It is clear that the neutral space
$\Cal N_t(S^{[a,b]}x)$ can be identified canonically
with $\Cal N_0(S^{[a,b]}x)$ by the usual identification of the
tangent spaces of $\boq$ along the trajectory $S^{(-\infty,\infty)}x$
(see, for instance, Section 2 of \kssb). As we will see in Section 3,
the neutral subspace is the orthocomplement of the positive
subspace of the second fundamental form of the image of a parallel beam
of light, see the proof of Proposition 3.18 later.
It is now time to bring up the basic notion of {\it sufficiency} of a
trajectory (segment). This is the utmost important necessary condition for
the proof of the Theorem on Local Ergodicity for Semi--Dispersing billiards,
see Condition (ii) of Theorem 3.6 and Definition 2.12 in \kssb.
\proclaim{Definition 2.3}
\roster
\item
The non--singular trajectory segment $S^{[a,b]}x$ ($a$ and $b$ are supposed
not to be moments of collision) is said to be {\it 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
is said to be {\it sufficient} if and only if both branches of this
trajectory segment are sufficient.
\endroster
\endproclaim
For the notion of trajectory branches see, for example,
the end of Section 2 in [Sim(1992)-I].
\proclaim{Definition 2.4}
The phase point $x\in \bold M$ with {\it at most one 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$ is sufficient.
\endproclaim
In the case of an orbit $S^{(-\infty,\infty)}x$ with exactly one
singularity, sufficiency requires that both branches of
$S^{(-\infty,\infty)}x$ be sufficient.
\bigskip \bigskip
\heading
3. Characterization of the Action of $\Cal G$
\endheading
\bigskip \bigskip
As has been said in the introduction, we consider cylindric billiards
with the configuration space
$\bold Q=\Bbb T^d\setminus\left(C_1\cup\dots\cup C_k\right)$, where
$\Bbb T^d=\Bbb R^d/\Bbb Z^d$ ($d\ge 2$) is the unit torus, the generator
space of the cylinder $C_i$ is $A_i\subset\Bbb R^d$, $L_i=A_i^{\perp}$,
and $\dim L_i\ge 2$. Recall that the linear subspace $A_i\subset\Bbb R^d$
is a lattice subspace of $\Bbb R^d$, so that it defines a subtorus
$T_i=A_i/\Bbb Z^d$ of $\Bbb T^d$.
For a cylinder $C_i$ ($i=1,\dots ,k$) consider the group $\Cal G_i$
consisting of all orthogonal transformations $U$ of $\Bbb R^d$ that
leave the points of the subspace $A_i$ fixed. Since the initial studies of the
close relationship between the ergodicity of cylindric billiards and the
rotation groups determined by the generator spaces (cf. Lemma 4.4
and Sublemma 4.5 in [S-Sz (1994)] and the role of these lemmas in the
proof of Main Lemma 4.1 there)
it has become apparent that the transitivity of the action
of the algebraic generate of these $\Cal G_i$'s on the sphere $\Bbb S^{d-1}$
is vital for proving ergodicity. We investigate the subgroup $\Cal G$
of $O(d)$ algebraically generated by all such groups $\Cal G_i$,
which is certainly an embedded Lie subgroup of $O(d)$.
Observe that every orbit $\Cal GV$ ($V\in\Bbb S^{d-1}$)
of the action of $\Cal G$ on the unit sphere $\Bbb S^{d-1}$ is certainly an
embedded, smooth submanifold of $\Bbb S^{d-1}$, for it is naturally
diffeomorphic to the homogeneous space that can be obtained by taking $\Cal G$
modulo the isotropy subgroup of $V$.
Being aware of our general, three-step strategy of proving ergodicity for
semi-dispersive billiards, it is clear that without the mentioned transitivity
there would not be a great chance to prove the hoped ergodicity for such
billiards. Later on, in Theorem 3.6, we will see that without the
transitivity of the $\Cal G$-action actually there must exist some very
simple first integrals of the flow, thus hindering the ergodicity.
At this point, however, it is already
easy to see that without that transitivity
the flow $\flow$ simply can not be hyperbolic. Indeed, hyperbolicity
just means that for $\mu-$almost every point $x\in M$ the relevant Lyapunov
exponents of the system are nonzero. On the other hand, if for an
$x\in M$, the relevant exponents do not vanish, then the point is
necessarily sufficient. By the following lemma, however, the existence of just
one sufficient point already implies the transitivity of the action of
$\Cal G$.
Denote by $\delta$ the maximum dimension of the orbits of the action
of $\Cal G$ on the unit sphere $\Bbb S^{d-1}$.
\medskip
\proclaim{Lemma 3.1} The action of $\Cal G$ on $\Bbb S^{d-1}$ is
transitive if and only if $\delta=d-1$, i. e. there is an open orbit.
\endproclaim
\subheading{Proof} Transitivity obviously implies $\delta=d-1$.
Suppose that the orbit $\Cal G V$ of some $V\in\Bbb S^{d-1}$ is an
open subset of the unit sphere $\Bbb S^{d-1}$. Then there exists an
$\epsilon_0>0$ such that $B(V;\,\epsilon_0)\subset\Cal GV$. Since
$\Cal G\subset O(d)$, we have that $B(gV;\,\epsilon_0)\subset\Cal GV$
for all $g\in\Cal G$. Using the connectedness of $\Bbb S^{d-1}$,
this is only possible if $\Cal GV=\Bbb S^{d-1}$. \qed
\medskip
First we put forward a simple observation regarding the (possibly existing)
dense orbits of the $\Cal G$-action.
\proclaim{Lemma 3.2} Suppose that the orbit $\Cal GV\subset\Bbb S^{d-1}$
is dense for some $V\in\Bbb S^{d-1}$. Then every orbit of the $\Cal G$-action
is also dense.
\endproclaim
\subheading{Proof} Denote by $\overline{\Cal G}\subset O(d)$ the closure of
the group $\Cal G$ in the orthogonal group $O(d)$. It is obvious that the
closure $\overline{\Cal GV}$ of the orbit $\Cal GV$ is precisely the orbit
$\overline{\Cal G}V$ of the $\overline{\Cal G}$-action on $\Bbb S^{d-1}$.
Thus, $\overline{\Cal GV}=\Bbb S^{d-1}$ means that the
$\overline{\Cal G}$-action is transitive, hence
$\overline{\Cal G}V'=\overline{\Cal GV'}=\Bbb S^{d-1}$ for every
$V'\in\Bbb S^{d-1}$. \qed
Let us observe that the existence of a single dense orbit implies the
irreducibility of the $\Cal G$-action on the space $\Bbb R^d$, for every
orbit $\Cal GV'$ ($V'\in\Bbb S^{d-1}$) is then dense in $\Bbb S^{d-1}$.
Thus, irreducibility of the $\Cal G$-action is a necessary condition for
the transitivity.
\medskip
Now we present another necessary condition for transitivity:
\proclaim{Definition of Orthogonal Non-Splitting Property, ONSP} We say that
the system of subspaces $L_1,\dots ,L_k$ has the Orthogonal Non-Splitting
Property (ONSP) iff there is no orthogonal splitting
$\Bbb R^d=B_1\oplus B_2$ with $\dim B_i>0$ and with the property
that for every $i=1,\dots ,k$ either $L_i\subset B_1$ or
$L_i\subset B_2$.
\endproclaim
\medskip
\proclaim{Lemma 3.3} The irreducibility of the $\Cal G$-action on $\Bbb R^d$
implies the ONSP.
\endproclaim
\subheading{Proof} If ONSP fails to hold, then the subspaces $B_i$ in the
splitting described above (with the property that for every $i=1,\dots ,k$
either $L_i\subset B_1$ or $L_i\subset B_2$) are $\Cal G$-invariant.
We note that in this case, in addition to the total kinetic energy
$\dfrac{1}{2}\Vert V\Vert^2$, the cylindric billiard clearly has further
first integrals (invariant quantities), namely, the kinetic energies
corresponding to the invariant norm square
$\left\Vert P_{B_i}(V)\right\Vert^2$ ($i=1,\, 2$)
of the orthogonal projection of the velocity $V$ on the
subspace $B_i$. \qed
\medskip
\proclaim{Lemma 3.4} The irreducibility of the action of
$\Cal G$ on $\Bbb R^{d}$ is equivalent to the ONSP.
\endproclaim
\subheading{Proof}
We have seen that the irreducibility implies the ONSP. Suppose
now the reducibility. i. e. the existence of an orthogonal splitting
$\Bbb R^d=B_1\oplus B_2$, $\dim B_i>0$, and $B_i$ is $\Cal G$-invariant.
We show that ONSP is violated by the same splitting $\Bbb R^d=B_1\oplus B_2$.
Consider and fix an arbitrary index $i$, $1\le i\le k$.
\medskip
\proclaim{Sublemma} If the linear subspace $B\subset\Bbb R^d$ is
$\Cal G_i$-invariant, then $B$ has the orthogonal direct sum splitting
$B=B^* \oplus B^{**}$, where $B^{**}\subset A_i$ and $B^*=0$ or
$B^*=L_i$. (Recall that $\Cal G_i\subset\Cal G$ consists of all
orthogonal transformations $U\in O(d)$ that leave all vectors of $A_i$
fixed.)
\endproclaim
\medskip
\subheading{Proof} If $B$ is a subspace of $A_i=L_i^\perp$, then we are done:
$B^{**}=B$, $B^*=0$. Assume now that the orthogonal projection
$P_{L_i}(B)$ is nonzero. Select a vector $b\in B$ with $P_{L_i}(b)\ne 0$.
Since the tangent space $\Cal T_b\left(\Cal G_ib\right)$ is equal to
the orthocomplement $L_i\ominus P_{L_i}(b)=H_i$, by the invariance of $B$
we have that $H_i\subset B$. Since $\Cal G_i$ acts transitively on the unit
sphere of $L_i$, it follows that $L_i\subset B$. Take $B^*=L_i$,
$B^{**}=B\ominus L_i$. \qed
\medskip
Consider now the splittings $B_1=B_1^*\oplus B_1^{**}$,
$B_2=B_2^*\oplus B_2^{**}$. Then we have that either $B_1^*=L_i$
or $B_2^*=L_i$, for $B_1\oplus B_2\not\subset A_i$. This shows that
either $L_i\subset B_1$ or $L_i\subset B_2$, $i=1,\dots ,k$. \qed
\medskip
\subheading{Remark 3.5} Each of the above properties obviously implies that
$\bigcap_{i=1}^k A_i=0$.
\medskip
It is now a bit surprising fact that the irreducibility of the $\Cal G$-action
(or, equivalently, the ONSP) in turn implies the transitivity of the action
on $\Bbb S^{d-1}$, thus giving us an easily checkable
necessary and sufficient condition for transitivity. The equivalence of
transitivity and irreducibility is, indeed, unexpected to some extent
because of the following fact:
Consider the case when the dimension of every space $L_i$ is one.
(Of course, this situation is not in the scope of our study.)
Then the action of the group generated by the reflections
across the hyperplanes $A_i=L_i^\perp$ ($i=1,\dots ,k$) can very well produce
dense (but, of course, countable) orbits on the sphere $\Bbb S^{d-1}$,
thus making the representation irreducible without transitivity on
$\Bbb S^{d-1}$. This somehow shows the essential difference between
the cases $\dim L_i>1$ and $\dim L_i=1$.
\medskip
\proclaim{Theorem 3.6} The irreducibility of the $\Cal G$-action on $\Bbb R^d$
implies the transitivity on $\Bbb S^{d-1}$, i. e. these properties are
actually equivalent.
\endproclaim
\medskip
\proclaim{Corollary A} The transitivity of the $\Cal G$-action on
$\Bbb S^{d-1}$ is a necessary condition for the ergodicity of the
cylindric billiard.
\endproclaim
\medskip
\subheading{Proof} Without transitivity there must exist a non-trivial
orthogonal splitting $\Bbb R^d=B_1\oplus B_2$, and the partial kinetic
energies $\dfrac{1}{2}\left\Vert P_{B_i}(V)\right\Vert^2$ are first
integrals of the motion. \qed
\medskip
\proclaim{Corollary B. The Orbit Structure} The natural representation
of the group $\Cal G$ in $\Bbb R^d$ is always completely reducible, i. e.
the space $\Bbb R^d$ splits into the orthogonal direct sum
$\Bbb R^d=\bigoplus_{j=1}^r B_j$, where each subspace $B_j$ is
$\Cal G$-invariant, and the restriction of the representation to $B_j$
is irreducible. Note that the above splitting is also unique, as long
as the subspaces $L_i$ span the whole space $\Bbb R^d$. Two subspaces
$L_i$ and $L_j$ belong to the same component $B_s$ ($s=1,\dots ,r$)
if and only if these subspaces can be connected by a finite chain of
spaces $L_{i(1)},L_{i(2)},\dots ,L_{i(p)}$ so that the consecutive
elements in this chain are not orthogonal to each other. It follows from
Theorem 3.6 that every orbit of the $\Cal G$-action on $\Bbb S^{d-1}$
has the following form:
$$
\left\{V\in\Bbb S^{d-1}\big|\; \left\Vert P_{B_j}(V)\right\Vert^2=2E_j,
\quad j=1,\dots ,r \right\}.
$$
Here the quantity $E_j$ is the flow invariant partial kinetic energy
in the direction of the subspace $B_j$, $j=1,\dots ,r$. The orbit is
uniquely determined by the vector $(E_1,E_2,\dots ,E_r)$ fulfilling
$E_j\ge 0$ and $\sum_{j=1}^r E_j=1/2$. Thus every orbit is compact,
for it is the product of spheres. The typical orbit is the topological
product of $(\nu_j-1)$-dimensional spheres, $j=1,\dots ,r$,
where $\nu_j=\dim B_j$.
\endproclaim
\medskip
\subheading{Proof of Theorem 3.6}
The proof will be split into several lemmas, some of them being just very
simple observations.
\medskip
\proclaim{Lemma 3.7} Let us introduce the following equivalence relation
$\sim_i$ ($i=1,\dots ,k$) between vectors of $\Bbb R^d$:
$$
\aligned
& V_1\sim_i V_2\text{ if and only if } \\
& V_1-P_iV_1=V_2-P_iV_2 \text{ and }
\Vert P_iV_1\Vert=\Vert P_iV_2\Vert.
\endaligned
\tag 3.8
$$
(Recall that $P_i$ denotes the orthogonal projection onto $L_i$.)
Suppose that some subgroups (not necessarily Lie subgroups)
$\Cal H_i\subset\Cal G_i=\text{O}(L_i)$ are given with the property that
$\Cal H_i$ acts transitively on the unit sphere of $L_i$, $i=1,\dots ,k$.
Then the algebraic generate $\Cal H=\langle\Cal H_1,\dots ,\Cal H_k\rangle$
of the groups $\Cal H_i$ acts transitively on $\Bbb S^{d-1}$ if and
only if the transitive hull of the equivalence relations $\sim_i$
on $\Bbb S^{d-1}$ ($i=1,\dots ,k$) is the trivial equivalence relation on
$\Bbb S^{d-1}$ making every pair of velocities equivalent.
\endproclaim
\subheading{Proof} The assertion of the lemma immediately follows from
the definitions. \qed
\medskip
Based on the previous lemma, it will be convenient to say that
the system of subspaces $\left\{L_1,\dots ,L_k\right\}$ of $\Bbb R^d$
($\dim L_i\ge 2$, $i=1,\dots ,k$) is {\it transitive} if and only if the
transitive hull of the relations $\sim_i$ ($i=1,\dots ,k$) associated with
the subspaces $L_1,\dots ,L_k$ is the trivial relation
$\Bbb S\times\Bbb S$ on the unit sphere $\Bbb S$ of the linear span
$L_1+L_2+\dots +L_k$ of these subspaces.
\medskip
\proclaim{Lemma 3.9} If the systems of linear subspaces
$\left\{L_1,L_2,\dots ,L_p\right\}$ and
$$
\left\{L_1+L_2+\dots +L_p,L_{p+1},\dots ,L_k\right\}
$$
are both transitive ($1\le p\le k$), then the system
$\left\{L_1,\dots ,L_k\right\}$ is transitive, as well.
\endproclaim
\subheading{Proof} The lemma is an immediate consequence of Lemma 3.7. \qed
\medskip
The next proposition deals with the transitivity of two subspaces
$\{L_1,L_2\}$ of $\Bbb R^d$.
\proclaim{Proposition 3.10} The system of two subspaces
$\{L_1,L_2\}$ of $\Bbb R^d$ ($\dim L_i\ge 2$, $i=1,\, 2$) is transitive if
and only if $L_1$ and $L_2$ are not orthogonal to each other.
\endproclaim
\subheading{Proof} Non-orthogonality is, of course, necessary for
transitivity. Assume now that $L_1\not\perp L_2$, and show that the system
$\{L_1,L_2\}$ is transitive. In order to simplify the notations, we assume
that $L_1+L_2=\Bbb R^d$.
\medskip
\subheading{Case I. $L_1\cap L_2\ne0$} Select a vector $V\in\Bbb S^{d-1}$
such that $V\not\in(L_1\cap L_2)^{\perp}$. It is easy to see that the tangent
space $\Cal T_V\Cal G_iV$ of the orbit $\Cal G_iV$ at $V$ ($i=1,\, 2$)
is the linear space $V^\perp\cap L_i$ where, as usual, $\Cal G_i$ denotes
the group of all orthogonal transformations of $\Bbb R^d$ that keep the
vectors of $L_i^\perp =A_i$ fixed. Elementary computation with the dimensions
shows that
$$
\aligned
& \dim\left[(V^\perp\cap L_1)+(V^\perp\cap L_2)\right]=
\dim L_1-1+\dim L_2-1-\left(\dim L_1\cap L_2-1\right) \\
& =\dim(L_1+L_2)-1=d-1.
\endaligned
$$
Due to Lemma 3.1, the system $\{L_1,L_2\}$ is indeed transitive in Case I.
\medskip
\subheading{Case II. $L_1\cap L_2=0$} In this case the above calculus
of dimensions provides
$$
\dim\left[(V^\perp\cap L_1)+(V^\perp\cap L_2)\right]=d-2,
$$
so we have by one less dimension than the needed $d-1$. Interestingly enough,
the missing one dimension of Case II and the trick to obtain transitivity of
the action by using Lie brackets already appeared in [K-S-Sz(1991)].
Actually, we apply now the
commutator method used in the proof of Sublemma 4.5 of [S-Sz(1994)]. Indeed,
we select two-dimensional subspaces $\tilde L_i\subset L_i$ ($i=1,\, 2$)
so that $\tilde L_1$ and $\tilde L_2$ are still not orthogonal to each other,
then we consider the linear span $H=\tilde L_1+\tilde L_2$. In the spirit of
Lemma 3.9, in order to prove the transitivity of the system $\{L_1,L_2\}$
it is enough to show that
(A) the system $\{\tilde L_1,\, \tilde L_2\}$ is transitive,
\noindent and
(B) the system $\{L_1,L_2,H\}$ is transitive.
\medskip
By the result of Case I, both systems $\{L_1+H,\,L_2\}$ and $\{L_1,\,H\}$ are
transitive and, therefore, according to Lemma 3.9, the statement of (B) is
indeed true.
The only outstanding task in the proof of Proposition 3.10 is to show (A).
In order to simplify the notations, we assume that
$\tilde L_1+\tilde L_2=\Bbb R^4$. We have arrived at the set-up of Sublemma
4.5 of [S-Sz(1994)], except that now we do not have the bit stronger
assumption $\tilde L_1\cap(\tilde L_2)^\perp=0$ but, rather, we only have
that $\tilde L_1\not\perp\tilde L_2$. This means that the cosine
$$
b=\min\left\{\Vert P_2(x)\Vert |\; x\in\tilde L_1,\quad ||x||=1\right\}
\tag 3.11
$$
of the maximum angle between a unit vector of $\tilde L_1$ and the plane
$\tilde L_2$ (see (4.6) in [S-Sz(1994)]) may be zero, whereas the cosine
$$
a=\max\left\{\Vert P_2(x)\Vert |\; x\in\tilde L_1,\quad ||x||=1\right\}
\tag 3.12
$$
of the minimum such angle (see (4.5) in [S-Sz(1994)]) must be strictly between
zero and one, thanks to the conditions $\tilde L_1\cap\tilde L_2=0$
and $\tilde L_1\not\perp\tilde L_2$.
However, the whole machinery of computing the commutator
$\left[X_1,X_2\right]$ of the infinitesimal generators $X_1,\, X_2$ of the
one-parameter rotation groups $\Cal G_1=O\left(\tilde L_1\right)$,
$\Cal G_2=O\left(\tilde L_2\right)$ (presented in the proof of Sublemma 4.5
of [S-Sz(1994)]) still works, and we obtain the matrix expansion
$$
[X_1,X_2]=\left(\matrix-ab&0&-b&0 \\
0&-ab&0&-a \\
b&0&ab&0 \\
0&a&0&ab\endmatrix\right)
\tag 3.13
$$
for the commutator $\left[X_1,X_2\right]$ written in an appropriate basis
$\{e_1,f_1,e_2,f_2\}$ of $\Bbb R^4$. Here $e_1\in\tilde L_1$ is a unit vector
(out of the two possible, antipodal unit vectors) for which the maximum
value $a$ of $\Vert P_2(x)\Vert$ is attained,
$e_2=P_2(e_1)/a\in\tilde L_2$, whereas $f_1\in\tilde L_1\ominus e_1$
is a unit vector (out of the two possible, antipodal ones)
for which the minimum value
$b$ of $\Vert P_2(x)\Vert$ is attained, and, finally,
$f_2\in\tilde L_2\ominus e_2$ is a unit vector making the angle
$\arccos(b)$ with $f_1$. Note that $\langle e_1,\, e_2\rangle=a$,
$\langle f_1,\, f_2\rangle=b$, and $\langle e_i,\, f_j\rangle=0$,
$i,\, j=1,\, 2$. Let us compute now the action of the infinitesimal
generators $X_1$, $X_2$, and $[X_1,\, X_2]$ on the nonzero vector $f_1+f_2$,
that is, multiply this vector from the left by the matrices of the above
infinitesimal generators. Formula (3.13) and the definition of the
infintesimal generators $X_i$ of the one-parameter rotation groups
$\Cal G_i=\text{O}(\tilde L_i)$ yield the following results:
$$
[X_1,\, X_2]\cdot(f_1+f_2)=a(b+1)(f_2-f_1),
$$
$$
X_1\cdot(f_1+f_2)=X_1\cdot\left(P_1(f_1+f_2)\right)=\pm(b+1)e_1,
$$
and, similarly,
$$
X_2\cdot(f_1+f_2)=\pm(b+1)e_2.
$$
The resulting three vectors are apparently linearly independent and,
therefore, the $\Cal G$-orbit of the vector $f_1+f_2$ (in the sphere of
radius $\Vert f_1+f_2\Vert$) is open. By Lemma 3.1, this implies the
transitivity of the $\Cal G$-action on the unit sphere $\Bbb S^3$.
This completes the proof of Proposition 3.10. \qed
\medskip
\subheading{Finishing the proof of Theorem 3.6} By using induction on the
number $k$ of the subspaces $L_1,L_2,\dots ,L_k$ ($\dim L_i\ge 2$), we prove
that the system $\{L_1,\dots ,L_k\}$ is transitive, as long as it has the
ONSP (see the definition preceding Lemma 3.3) in the linear span
$L_1+L_2+\dots +L_k$.
Indeed, the statement is obviously true for $k=1$. Let $k>1$, and assume that
the theorem has been proven for all smaller values of $k$. By the ONSP of
the system $\{L_1,\dots ,L_k\}$, one can find two subspaces among
$L_1,\dots ,L_k$, say $L_1$ and $L_2$, such that $L_1\not\perp L_2$.
According to Proposition 3.10, the system $\{L_1,L_2\}$ is transitive.
The ONSP of the collection $\{L_1,\dots ,L_k\}$ immediately implies the ONSP
for the system $\{L_1+L_2,L_3,\dots ,L_k\}$. By the induction hypothesis,
the latter system is transitive. Finally, Lemma 3.9 and the mentioned
transitivity of $\{L_1,L_2\}$ yield the transitivity of
$\{L_1,\dots ,L_k\}$. Theorem 3.6 is now proved. \qed
\medskip
\subheading{Remark 3.14} The validity of the following assertion easily
follows from the proof of Lemma 3.7, actually from the characterization
of the transitivity in terms of the equivalence relations $\sim_i$,
$i=1,\dots ,k$:
If the system $\{L_1,L_2,\dots ,L_k\}$ ($\dim L_i\ge 2$,
$L_1+\dots +L_k=\Bbb R^d$) is transitive, then even the connected
component $\Cal G^0$ of the unity in the Lie group $\Cal G$ acts
transitively on the unit sphere $\Bbb S^{d-1}$. Note that $\Cal G^0$
is the group algebraically generated by the connected components
$\text{SO}(L_i)$ of the groups $\Cal G_i=\text{O}(L_i)$, $i=1,\dots ,k$.
\medskip
It is worth mentioning the following, easily checkable sufficient condition
for transitivity:
\proclaim{Definition of tightness} We say that the system of subspaces
$L_1,\dots ,L_k$ is tight if and only if there exists a system of $d-1$
linearly independent vectors
$e_1\in L_{i(1)},\, e_2\in L_{i(2)},\, \dots ,\, e_{d-1}\in L_{i(d-1)}$
such that the linear span $\text{span}\left\{e_1,\dots ,e_{d-1}\right\}$
of these vectors does not contain any of the subspaces
$L_{i(1)},\dots ,L_{i(d-1)}$.
\endproclaim
As a matter of fact, as we will see in the proof of the forthcoming lemma,
under the condition of tightness, transitivity can be obtained directly
by merely taking the linear span of the Lie algebras of $\Cal G_i$
instead of the entire generated Lie algebra.
\medskip
\proclaim{Lemma 3.15} The tightness of the system $L_1,\dots ,L_k$ implies
the transitivity of the action of $\Cal G$ on $\Bbb S^{d-1}$.
\endproclaim
\subheading{Proof} Select a system of linearly independent vectors
$e_1,\dots ,e_{d-1}$ featuring the definition of tightness. Let
$V\in\Bbb S^{d-1}$ be orthogonal to all vectors
$e_1,\dots ,e_{d-1}$. Consider one of these vectors, say, $e_1$.
Let us denote by $P_{i(1)}$ the orthogonal projection onto the
subspace $L_{i(1)}$. The projection $P_{i(1)}V$ is nonzero,
for $L_{i(1)}\not\subset\text{span}\left\{e_1,\dots ,e_{d-1}\right\}$.
Since the tangent space $\Cal T_V\left(\Cal G_{i(1)}V\right)$
of the orbit $\Cal G_{i(1)}V\subset \Cal GV$ at $V$ is the
orthocomplement $L_{i(1)}\ominus\text{span}\left\{P_{i(1)}V\right\}$
of the nonzero vector $P_{i(1)}V$ in $L_{i(1)}$,
we have that $e_1\in\Cal T_V\left(\Cal G_{i(1)}V\right)\subset
\Cal T_V\left(\Cal GV\right)$. By the same argument,
$e_j\in\Cal T_V\left(\Cal GV\right)$ for $j=1,\dots ,d-1$ and, therefore,
$\dim\left(\Cal GV\right)=d-1$. \qed
Basic examples related to the notion of tightness are given in Remarks
3.24 and 3.26. The first of them shows that transitivity does not imply
tightness, whereas the second one provides an example of a tight
family.
\bigskip \medskip
\heading
Characterization of the Positive Subspace of the Second Fundamental Form
\endheading
\bigskip
We define the ``parallel beam of light'' $B$ (formerly called an
orthogonal manifold) around the phase point $x_0=(Q_0,V_0)\in\bold{M}$ as
follows:
$$
B=\left\{x=(Q,V_0)\in\bold{M}:\; Q-Q_0\perp V_0 \text{ and } ||Q-Q_0||<
\varepsilon_0\right\}
\tag 3.16
$$
with a fixed and sufficiently small number $\varepsilon_0>0$. There are two
interpretations of the manifold $B$:
(a) a ($d-1$)-dimensional submanifold of the phase space $\bold{M}$, or
(b) a codimension-one flat submanifold $\pi(B)$ of the configuration space
$\bold{Q}$ supplied with a field of unit normal vectors, where
$\pi:\; \bold{M}\to\bold{Q}$ is the natural projection.
We shall use these two interpretations alternately: when dealing with the
second fundamental form we use (b), however, when
defining the image $S^t(B)$ of $B$ under the flow, we use the first
interpretation. Consider a non-singular trajectory segment $\traj$
and denote by $\Cal W_+=\Cal W_+\left(\traj\right)$ the
positive subspace of the positive semi-definite, symmetric, second
fundamental form $W$ of $S^T(B)$ at the point $S^Tx_0=(Q'_0,V'_0)$.
Since $W$ acts in the orthocomplement $(V'_0)^{\perp}$ of $V'_0$ in
$\Bbb R^d$, $\Cal W_+$ is a subspace of $(V'_0)^{\perp}$.
With the first collision with the cylinder $C_{\sigma_1}$ we associate a
($\nu_1-1$)-dimensional real projective space
$\Cal P_1\cong\Bbb P^{\nu_1-1}(\Bbb R)$ of all orthogonal
reflections of the space $\Bbb R^d$ across all possible hyperplanes
$H$ that contain the generator space $A_{\sigma_1}$ of the cylinder
$C_{\sigma_1}$. Plainly, such reflections (or hyperplanes $H$) can be
characterized uniquely by the orthocomplement line
$H^{\perp}\subset L_{\sigma_1}$, so the collection $\Cal P_1$ of all
such reflections is naturally diffeomorphic to the real projective
space $\Bbb P^{\nu_1-1}(\Bbb R)$, where $\nu_1=\dim L_{\sigma_1}$.
Similarly, other real projective spaces
$\Cal P_i\cong\Bbb P^{\nu_i-1}(\Bbb R)$ are attached to the
symbolic collisions $\sigma_i$, $i=1,\dots ,m$. By using these
definitions, we obtain a mapping
$$
\Phi_{\Sigma}=\Phi:\quad \sphere\times\szorzat\to\sphere
\tag 3.17
$$
which assignes to every ($m+1$)-tuple
$$
(V;h_1,h_2,\dots ,h_m)\in\sphere\times\szorzat
$$
the image $Vh_1h_2\cdot\dots\cdot h_m=V'$ of $V\in\sphere$ under the composite
action $h_1h_2\cdot\dots\cdot h_m$. (Here, by convention, $h_1$ is applied
first.) Plainly $V'_0=\Phi(V_0;g_1,g_2,\dots ,g_m)$, where $g_i$ denotes the
orthogonal reflection that causes the abrupt change of the velocity at the
$i$-th collision $\sigma_i$ of the given orbit segment $\traj$. The space
$\sphere\times\szorzat$
will often be called the phase space of the virtual dynamics, or the
phase space of the velocity process.
Now we can consider the partial derivative
$\dfrac{\partial\Phi}{\partial\tilde{\Cal P}}\pont$
of $\Phi$ with respect to the factor
$\tilde{\Cal P}=\szorzat$: It is a linear mapping from
the tangent space $\Cal T_{\vec{h}}\left(\tilde{\Cal P}\right)$
into the tangent space
$\Cal T_{V'}\sphere$, where $V'=\Phi\pont$. The next result characterizes the
positive subspace $\Cal W_+$ as the range of the above mentioned partial
derivative:
\medskip
\proclaim{Proposition 3.18} Using the definitions and notations from above,
$$
\Cal W_+\left(\traj\right)=\text{Ran}\left(\dfrac{\partial\Phi}
{\partial\tilde{\Cal P}}\pontg\right).
$$
\endproclaim
\subheading{Proof} The left-hand-side is obviously a subspace of the one
on the right. Conversely, suppose a vector
$$
Y=(y_1,y_2,\dots ,y_N)\in\left(V'_0\right)^{\perp}
$$
is orthogonal to $\Cal W_+$. We will show that $Y$ is orthogonal to
$$
\text{Ran}\left(\dfrac{\partial\Phi}{\partial\tilde{\Cal P}}\pontg\right),
$$
as well. Because of
$$
\aligned
& \text{Ran}\left(\dfrac{\partial\Phi}
{\partial\tilde{\Cal P}}\pontg\right)= \\
& \Cal L\left\{\text{Ran}\left(\dfrac{\partial\Phi}
{\partial\Cal P_i}\pontg\right):\; i=1, \dots,m\right\}
\endaligned
$$
it is sufficient to show that for every integer $i$ ($1\le i\le m$)
$$
Y\perp\text{Ran}\left(\dfrac{\partial\Phi}{\partial\Cal P_i}
\pontg\right):=R_i.
$$
For the sake of simplifying the notations we suppose that
$\sigma_i=1$. Then
$$
R_i=\left\{zg_{i+1}\dots g_m:\; z\in L_1,\; z\perp V^+_i \right\},
$$
where $V_i^+$ is the velocity right after the $i$-th collision with
the cylinder $C_1$. By the assumed neutrality
of $Y$ with respect to $\traj$ (which says that the vector
$Yg_m^{-1}\dots g_{i+1}^{-1}$ belongs to the linear span of
$A_1=A_{\sigma_i}$ and $V_i^+$), we have that
$Yg_m^{-1}\dots g_{i+1}^{-1}\perp z$ for every vector $z\in L_1$,
$z\perp V_i^+$. Thus, by the orthogonality of the mapping
$g_{i+1}\dots g_m$, we have $Y\perp R_i$.
Hence Proposition 3.18 follows. \qed
\medskip
\subheading{\bf Remark 3.19} The big advantage of the above lemma is that
it gives us a new characterization of sufficiency in terms of the pure
velocity process without the configuration history.
\bigskip
Proposition 3.18 pregnantly shows an
intimate relationship between the sufficiency
of a trajectory segment $\traj$ (i. e. when the left-hand-side in the
statement of that lemma has the maximum dimension $d-1$) and the transitivity
of the action of $\Cal G$ on the velocity sphere $\Bbb S^{d-1}$. This
circumstance and the results of the papers [S-Sz(1994)], [S-Sz(1995)],
and [Sim(1998)] (Especially Lemma 4.4 and Sublemma 4.5 in [S-Sz (1994)],
and the role of those lemmas in the proof of Main Lemma 4.1
in [S-Sz(1994)]) strongly suggest the following conjectures:
\medskip
\proclaim{Conjecture 1} For every cylindric billiard flow $\flow$
the transitivity of the action of $\Cal G$ on the velocity sphere
$\Bbb S^{d-1}$ is equivalent to the hyperbolic ergodicity (or, equivalently,
to the hyperbolicity and Bernoulli property; cf. [Ch-H(1996)] and
[O-W(1996)]) of the billiard map.
\endproclaim
\medskip
\subheading{Corollary 3.20. Dichotomy} If the above conjecture holds true,
then for every cylindric billiard flow $\flow$ exactly one of the following
two possibilities will occur:
\medskip
(I) The system $\{L_1,\dots ,L_k\}$ of the orthocomplements of the generators
has the ONSP in $\Bbb R^d$, and the billiard map is hyperbolic and enjoys
the Bernoulli property;
\medskip
(II) The system $\{L_1,\dots ,L_k\}$ has an orthogonal splitting
$\Bbb R^d=B_1\oplus B_2$ (see the definition right before Lemma 3.3),
and the partial kinetic energies $\left\Vert P_{B_1}(V)\right\Vert^2$
and $\left\Vert P_{B_2}(V)\right\Vert^2$ are trivial first integrals
of the motion.
\medskip
The above dichotomy shows that non-ergodicity can only be caused by
the presence of some very simple invariant quantity, namely the kinetic
energy of a ``subsystem''.
\medskip
\subheading{Corollary 3.21} Another consequence of Conjecture 1 is that
``the more cylinders the better'', i. e. (hyperbolic) ergodicity can not
be spoilt by the removal of additional cylinders from the configuration
space, that is, by adjoining more cylindric scatterers to the billiard.
\medskip
\subheading{Corollary 3.22} If Conjecture 1 is valid, then the phenomena of
ergodicity and partial hyperbolicity (i. e. the case when there exist
zero and nonzero Lyapunov exponents alike) can not coexist in cylindric
billiards.
\medskip
It is interesting to note that for billiards with two cylindric scatterers
the recent generalization of the result of [S-Sz(1994)] by P\'eter
B\'alint [B(1998)] actually verifies Conjecture 1 for that case.
\medskip
For completeness, we recite here the conjecture appeared in [Sz(1996)]
which easily turns out to be a weakened version of the previous
conjecture:
\proclaim{Conjecture 2} For every cylindric billiard flow $\flow$
the existence of a single sufficient trajectory is equivalent to the
hyperbolic ergodicity (or, equivalently, to the hyperbolicity and Bernoulli
property) of the billiard map.
\endproclaim
\medskip
\subheading{Remark 3.23} By virtue of Proposition 3.18
it is obvious that the existence of a sufficient
trajectory implies the transitivity of the $\Cal G$-action. Thus
Conjecture 1 is formally stronger than Conjecture 2.
\medskip
\subheading{Remark 3.24} Transitivity does not imply tightness. Indeed,
in the model of [S-Sz(1994)] we had $d=4$, and $\Bbb R^4$ was the linear
direct sum of the two-dimensional subspaces $L_1$ and $L_2$. Thus, for any
system $e_1\in L_{i(1)},\, e_2\in L_{i(2)},\, e_3\in L_{i(3)}$
of linearly independent vectors either $L_1$ or $L_2$ is a subspace of
$\text{span}\left\{e_1,e_2,e_3\right\}$, and the system
$\left\{L_1,\, L_2\right\}$ is not tight, although,
as has been shown in [S-Sz(1994)], the $\Cal G$-action is still transitive.
\medskip
\subheading{Remark 3.25. Orthogonal Cylindric Billiards} In his paper
[Sz(1994)] the second author studied the so-called orthogonal cylindric
billiards. i. e. the cylindric billiards for which each space
$L_i\subset\Bbb R^d$ is spanned by the coordinate axes belonging to the
set $K^i\subset\{1,2,\dots ,d\}$. The {\it sufficient and necessary condition
of ergodicity} found there was the following one: There is no splitting
$\{1,2,\dots ,d\}=B_1\cup B_2$, $B_j\ne\emptyset$,
$B_1\cap B_2=\emptyset$, such that every set $K^i$ is the subset of $B_1$
or $B_2$. It is clear that this condition is equivalent to our ONSP.
An interesting, special family of the above orthogonal cylindric billiards
is the one for which $\bigcap_{i=1}^k K^i\ne\emptyset$,
$\bigcup_{i=1}^k K^i=\{1,2,\dots ,d\}$. For simplicity assume that
$d\in \bigcap_{i=1}^k K^i$. Then we can select the standard coordinate
unit vectors $e_1,\dots ,e_{d-1}$ (in the direction of the first, second,
..., $(d-1)$-th coordinate axes), and this system obviously fulfills all
requirements in the definition of tightness. By the result
of [Sz(1994)], the corresponding cylindric billiard map is hyperbolic and
ergodic.
\medskip
\subheading{Remark 3.26} Consider the case when $\dim A_j=1$ for
$j=1,\dots ,k$. Assume that not all lines $A_j$ are parallel, for
example, $A_1$ is not parallel to $A_2$. Choose a linear basis
$e_1,\, \dots ,\, e_d$ in $\Bbb R^d$ so that
$\left\{e_1,e_3,\dots ,e_d\right\}$ is a basis of $L_1$ and
$\left\{e_2,e_3,\dots ,e_d\right\}$ is a basis of $L_2$.
(Thus, $\left\{e_3,\dots ,e_d\right\}$ is automatically a basis of
$L_1\cap L_2$.) Then the system $\left\{e_1,e_2,\dots ,e_{d-1}\right\}$
obviously fulfills all requirements in the definition of tightness.
Thus, we obtained that any cylindric billiard with one-dimensional
generators is necessarily tight, unless all generators are parallel.
(In which case the $\Cal G$-action is clearly not transitive.)
For such models the proof of (hyperbolicity and)
ergodicity can go ahead straightforwardly
along the lines of the proof of ergodicity developed in the papers
[K-S-Sz(1989)] and [S-Sz(1995)] for two cylinders. Thus, for the case of
one-dimensional generator spaces, the methods of [K-S-Sz(1989)] and
[S-Sz(1995)] actually prove Conjecture 1.
\bigskip \bigskip
\heading
4. Hard Sphere Systems
\endheading
\bigskip \bigskip
Consider the system of $N$ ($\ge 2$) hard spheres, labelled by $1,2,\dots ,N$,
with positive masses $m_1,\dots ,m_N$ ($\sum_{i=1}^N m_i=1$) moving uniformly
and colliding elastically in the unit $\nu$-torus ($\nu\ge 2$)
$\Bbb T^\nu=\Bbb R^\nu/\Bbb Z^\nu$. For simplicity we assume that these
spheres have the common radius $r>0$, so that even the interior of the
configuration space is connected. We make the standard reductions
$I=\sum_{i=1}^N m_iv_i=0$, $2E=\sum_{i=1}^N m_i\Vert v_i\Vert^2=1$,
where $v_i=\dot q_i$ is the velocity of the $i$-th sphere, while
$q_i\in\Bbb T^\nu$ is the position of its center. Corresponding to the
reduction $I=0$, we need to factorize the configuration space with respect
to uniform translations as follows: the configuration
$(q_1,q_2,\dots ,q_N)$ is considered to be equivalent to another
configuration $(q'_1,q'_2,\dots ,q'_N)$ iff there is an element
$a\in\Bbb T^\nu$ such that $q_i-q'_i=a$ for every index $i$. After this
factorization the configuration space $\bold Q$ is still a torus
(of dimension $d=\nu(N-1)$) from which we remove the convex sets
$$
C_{i,j}=\left\{(q_1,\dots ,q_N)\big|\; \Vert q_i-q_j\Vert_e<2r \right\},
$$
$1\le i0\}
$$
has Liouville measure zero, which is a direct consequence of the
invariance of the Liouville measure. Then we can discard the zero
measure union of the above sets, and this minor modification of the phase
space does not influence in any way the hyperbolicity status of the hard
sphere system.
\medskip
The result of this section is
\proclaim{Proposition 4.9} In the case of the hard sphere system
$(N;\nu;r;m_1,m_2,\dots ,m_N)$ the action of $\Cal G$ on $\Bbb S^{d-1}$
is transitive.
\endproclaim
\subheading{Proof} By using (4.4), we will simply check the meaning of the
orthogonality of two subspaces $L_{i,j}$ and $L_{k,l}$
($1\le i>1$ (very big), $m_3=m_4=\dots =m_N=1$, $v_1=v_2=0$, $N\ge 4$.
By the theorem and the previous remark this velocity configuration can be
transformed into $(V,\, -V,\, 0,\, 0,\,\dots ,\, 0)$ (where
$\Vert V\Vert^2=1\big/(2M)$) by applying at most $C$ elementary
transformations. We can, however, easily estimate (from above) the
maximum amount of kinetic energy that can be conveyed from the
subsystem $\{3,4,\dots ,N\}$ to the system $\{1,2\}$. Assume, for instance,
that the first and third spheres collide. It is easy to see that the maximum
amount of such an energy exchange occurs when
(a) the collision is a ``head-on'' collision, i. e. the two spheres
move on a straight line so that (by identifying this line with
$\Bbb R$ by also suitably orienting it) the pre-collision velocities
of the colliding spheres are $v^-_3>v^-_1>0$;
(b) the velocities $v^-_1,\, v^-_3$ have the maximum possible values.
\noindent
The simple reason for (b) to hold is that by increasing either $v^-_1$ or
$v^-_3$ the amount of energy gained by the first particle during this
collision also increases.
Straightforward upper bounds for these positive velocities are
$v^-_1\le 1/\sqrt{M}$, $v^-_3\le 1$. An elementary calculation
shows that in the extreme case $v^-_1=1/\sqrt{M}$, $v^-_3=1$
the after-collision velocity $v_3^+$ is equal to
$$
\frac{2\sqrt{M}+1-M}{M+1}
$$
and, therefore, the maximum order of magnitude of the amount of
kinetic energy conveyed from the subsystem $\{3,4,\dots ,N\}$ to
the subsystem $\{1,2\}$ (by a single collision)
is not greater than $2/\sqrt{M}$. Thus, in order for the
subsystem $\{3,\dots ,N\}$ to lose its energy $1/2$, there must be
at least $\sqrt{M}/4$ collisions between the two subsystems of
heavy and light particles, i. e.
$C(N;\nu;M,M,1,1,\dots ,1)\ge\sqrt{M}/4$, $M>>1$.
\medskip
\subheading{Remark 4.12. Arbitrary collision graphs} Suppose that not all
cylinders $C_{i,j}$ are removed from the original, $\nu(N-1)$-dimensional
configuration torus, but, instead, only the cylinders whose pair of labels
$\{i,j\}$ belongs to the set of edges $E(G)$ of a prescribed graph $G$ of
allowed collisions on the set of vertices $\{1,2,\dots ,N\}$. Then the proof
of Proposition 4.9 yields that the $\Cal G$-action corresponding to the
arising cylindric billiard is transitive if and only if the graph $G$ is
connected on the whole vertex set $\{1,2,\dots ,N\}$. Hyperbolic ergodicity,
that is, actually Conjecture 1 was proved to be true in [S-Sz(1995)]
for such classes of generalized hard sphere systems in the case
$\nu\ge4$, provided that the connected graph $G$ is either the simple path
of length $N-1$, or the simple loop of length $N$. For such graphs of
allowed collision, in the case of $\nu=3$, the complete hyperbolicity of
the flow is proved there, too.
Finally, by taking into account (4.3), observe the interesting fact that,
in the realm of such generalized hard sphere systems, the condition
$\bigcap_{(i,j)\in E(G)} A_{i,j}=0$
is also equivalent to the connectedness of the graph $G$. Thus, at least for
these generalized hard sphere systems, the transitivity and the otherwise
much weaker property $\bigcap_{i,j}A_{i,j}=0$ are equivalent.
\bigskip
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\enddocument
**