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\document
{\catcode`\@=11\gdef\logo@{}}
\vskip2cm
\centerline{\bf The K-property of `Orthogonal' Cylindric Billiards}
\vskip1cm
\centerline{Domokos Sz\'asz\footnote{Research
supported by the Hungarian National Foundation for
Scientific Research, grant No. 1902}}
\hbox{}
\centerline{Mathematical Institute of the Hungarian Academy of Sciences}
\centerline{H-1364, Budapest, P. O. B. 127, Hungary}
\vskip1cm
\hbox{\centerline{\vbox{\hsize8cm
\noindent
{\bf Abstract.}
Toric billiards with cylindric scatterers (briefly cylindric
billiards) generalize the class
of Hamiltonian systems of elastic hard balls.
In this paper a class of cylindric
billiards is considered where the cylinders are `orthogonal' or
more
exactly: the constituent space of any cylindric scatterer is spanned by
some of the (of course, orthogonal) coordinate vectors adapted to the
euclidean
torus. It is shown that the natural necessary condition for the
K-property of such billiards is also sufficient.
}}}
\vskip1cm
\bigskip
\heading 1. Introduction. Cylindric billiards.
\endheading
\medskip
It is well-known that Hamiltonian systems of elastic hard balls are
isomorphic
to certain billiards with cylindric scatterers (cf. [SCh-87],
[KSSz-91]).
Cylindric billiards, i. e. toric billiards with cylinders as scatterers,
belong to the class of semi-dispersing billiards (cf. [SCh-87]). They
deserve special attention
for they are relatively simple. Indeed, the first semi-dispersing but
not dispersing billiard
whose ergodicity was shown was a cylindric billiard introduced in
[KSSz-89] and the methods developed there for establishing global
ergodicity were quite instructive for more complicated models as well.
Consider compact affine subspaces $L^i: 1 \le i \le N$, $N \ge 1$ in the
$d$-torus $\bold {T}^d$
(with $ \text {\rm dim}\ L^i \le d-2$), and denote $C^i:=
\{ Q:= (q_1, \dots , q_d): \text {\rm dist} (Q, L^i)
\le r^i\},\ \ 1\le i \le N$ where each $r^i >0$. Denote $\Q := \bold
{T}^d \setminus (\cup _{i=1}^N C^i)$ and $M:= \Q \times
\bold {S}_{d-1}$ where $\bold {S}_{d}$ is the $d$-sphere. $M= \{x=
(Q,V)\}$ is
the phase-space of the billiard given in the domain $\Q$ with cylindric
scatterers. The dynamical system $(M, S^{\Bbb R}, d\mu)$, where
$S^{\Bbb R}$
is the dynamics defined
by uniform motion inside the domain and specular reflections at its
boundary (the scatterers!) and $d\mu$ is the Liouville measure, is
called a
{\it cylindric billiard}. (As to notions and notations in connection
with semi-dispersing billiards we follow the work [KSSz-90].)
In 1991 John Mather [M-91] asked whether a condition that, for
instance, the
intersection of the constituent subspaces of these cylinders is trivial
is sufficient to ensure the ergodicity of the corresponding billiard
(in connected components of its phase space, of course). I expect that
the answer is yes if, in addition, we
exclude the existence of trivial integrals of motion (a natural further
assumption is, of course, that the configuration space $\Q$ be connected
since otherwise each connected component should belong to different
ergonents). To illustrate the appearance of trivial
integrals of motion, though the intersection of the constituent
subspaces
is the zero-space, consider the billiard on the $4-$torus
with two cylindric
scatterers: $C^1:= \{Q \in \bold {T}^4: ( q_1^2 +q_2^2 )^{1/2} \le
r\}$ and
$C^2:= \{Q \in \bold {T}^4: ( q_3^2 +q_4^2 )^{1/2}\le r\}$. Here,
beside
the total energy, $v_1^2 + v_2^2$ and $v_3^2 +v_4^2$ are also
conserved and the billiard is, of course, not ergodic.
The conjecture in such a generality is very strong since it contains the
statement of the Boltzmann-Sinai ergodic hypothesis: the system of $N
\ge 2$ billiard balls on $\bold {T}^\nu, \nu \ge 2$ is ergodic
on connected components of the submanifold of the phase space specified
by the trivial integrals
of motion (energy, trajectory of the center of mass). This hypothesis is
still not settled in general; by recent beautiful results of Sim\'anyi
[S-92] it is only established for the case $\nu \ge N$ (beside the cases
settled earlier:
(i) $N=2,\ \ \nu \ge 2$, in [SCh-87] and also [S-70], [BS-73]; (ii)
$N=3,\
\ \nu \ge 2$, in [KSSz-91]; (iii) $N=4,\ \ \nu \ge 3$, in [KSSz-92];
we also note that the K-property of a special Hamiltonian system of
hard balls was obtained in [BLPS-92]).
Nevertheless we think that the class of cylindric billiards is interesting
not only because it contains hard ball systems. Semi-dispersing billiards, in
general, is a too wide and at the same time too wild category
whereas for cylindric biliards it is a realistic aim to obtain
transparent necessary and sufficient conditions.
In this paper we consider a special class of cylindric billiards: the
constituent space of any cylindric scatterer is spanned by some of the
coordinate vectors adapted to the orthogonal coordinate system where
$\bold {T}^d$ is given. We call these billiards {\it orthogonal
cylindric billiards}. In technical terms these billiards
are given by a family $C^j: 1 \le j \le J$ of cylinders
$C^j:=
\sigma_{u^j}\{(q_1, \dots , q_d): (\sum_{i \in K^j} q_i^2)^{1/2} \le
r^j\}$ on the
$d-$torus where
$\sigma_u$ denotes the shift with a vector $u \in
\bold {T}^d$. To concentrate to the essential part of our ideas {\it we
assume that the cylinders are disjoint} and later, at the end of Section 2,
we will indicate how this condition can be weakened.
Our main result (announced in [Sz-93]) is the following
\proclaim
{Theorem} Assume that, for an orthogonal cylindric billiard with
disjoint scatterers,
$\Q$ is a connected domain. Then the system is a K-system
if and only if $\{K^j: 1 \le j \le J\}$ is a connected family of
subsets of $\{1, \dots, d\}$.
\endproclaim
The proof is based on the strategy formulated in [KSSz-89] and
[KSSz-92].
Indeed, in the spirit of the latter work, the proof of global ergodicity
of a semi-dispersing billiard should be based on a suitable
definition of richness and then essentially consist of three parts:
\roster
\item geometric-algebraic part for treating neighbourhoods of rich
points,
\item dynamical-topological part for handling the subset of non-rich
points,
\item and, finally, separate arguments for singular trajectories (also
settling the Chernov-Sinai Ansatz).
\endroster
Our concept of richness, in its spirit, is not very far from that used
in [KSSz-92] and in [S-92].
Nonetheless, our method for proving part 1
is novel and is, in fact, quite instructive: it makes really clear why
the requirement of richness ensures the necessary codimension two
property of non-sufficient points.
We hope that its idea can be extended to a wider class of models as well.
It is worth noting that
in [KSSz-92] and [S-92], where hard ball systems are discussed,
this part of the proof requires the unpleasant assumption that the
dimension of the torus be not too small.
In part 2, the failure of the analogue of the Theorem of the Appendix of
[KSSz-92] requires a new approach. Nevertheless, in the proof of Lemma
4.7, we strongly rely on the proof given in [KSSz-92],
and will only present the details where they are different.
Finally, the proof given in [KSSz-92] for part 3, in principle, also
settles our case. Therefore we omit the proof, and, at the end of
section 4, we just put down some calculations to convince the reader
that, in fact, this is the case.
The paper is organized as follows:
in section 2 we present the notion of richness, formulate the Main
Lemmas and indicate, how they imply the theorem. Section 3 is devoted to
the geometric-algebraic part 1 by proving Main Lemma 2.2. Section 4
then settles the dynamical-topological part 2 by proving Main Lemma 2.3
and also contains the necessary remarks as to part 3. According to the
traditions, we also need some facts from topological dimension theory
that are summarized in Appendix 1. Appendix 2 is devoted to an
elementary analysis of trivial non-sufficient point of some natural
sub-billiards.
For brevity, our exposition relies heavily on earlier works:
first of all on [KSSz-92] and [KSSz-89] but also on [KSSz-90] and
[KSSz-91]. To help the reader, however, we everywhere provide precise
references
to the occurence of the necessary definitions, statements and
arguments.
\vskip2cm
\heading 2. Notion of richness, Main Lemmas and the proof.
\endheading
\medskip
Likewise as for hard ball systems, the proof is inductive but here
induction is only used for brevity of exposition. For
$d=2$,
a billiard satisfying the conditions of the Theorem is necessarily
dispersing and thus a K-system. Assume now that any billiard in
dimension $d' < d$ fulfilling the requirements of the Theorem possesses
the K-property. Under this condition we will show that the same holds
true in dimension $d$ as well. (Though we will have several inductive
arguments, the aforeformulated inductive assumption will, in fact, be
only used in the proof of Main Lemma 2.3. )
Similarly as in
[KSSz-91],
$M^*$ will denote the set of phase points whose orbits contain an
infinite number of collisions among which not more than one is singular.
$M^0 \subset M^*$ will be the subset of regular phase points, and
$M^1:= M^* \setminus M^0.$
Moreover, $\sr$ will denote the collection of all phase points
$x\in \partial M$ for which the reflection, occurring at $x$, is {\rm
singular}
(tangential or multiple) and, in the case of a multiple collision, $x$ is
supplied with the {\it outgoing} velocity $V^+$.
We remind the reader that a trajectory segment $S^{[a,b]}x$ is called
{\it regular} (or non-singular) if it does not hit singularities (
$S^{[a,b]}x \cap \sr = \emptyset$; cf. [KSSz-92]).
Consider the regular trajectory segment $S^{[a,b]}x,\ -\infty \le a \le
b \le \infty, x \in M$. Its symbolic collision
sequence is the list of subsequent cylinders of collisions $(C^{j_1},
\dots, C^{j_k}),\ \ 1 \le k\ $ of the
trajectory and can be described by the sequence $(j_1, \dots, j_k),\ \ 1
\le j_l \le J, 1 \le l \le k $. (If
the trajectory hits one or more singularities, then, of course, there
are a finite number of such sequences for any finite orbit.)
In the previous section we characterized a cylinder $C$ by a triple
$(K,r,u):\ K
\subset \{1, \dots, d\},\ r \in \Bbb R, u \in \bold {T}^d$.
\proclaim {Definition 2.1} We say that the trajectory segment $S^{[a,b]}x$ is
{\it connected} if $(K^{j_1}, \dots, K^{j_k})$ is a connected cover of
the set $\{1, \dots, d\}$. We say that the trajectory segment $S^{[a,b]}x$
is {\it rich} if there exists a time $t \in [a,b]$\ \ (with $S^tx \in
\partial M $ also allowed) such that both
trajectory segments $S^{[a,t]}x$ and $S^{[t,b]}x$ are connected.
If the trajectory segment hits singularities, then the
above properties are required for any trajectory branch.
Finally, the trajectory segment is {\it poor} if it not rich.
\endproclaim
\proclaim {Main Lemma 2.2} Assume that the trajectory segment
$S^{[a,b]}x$
is regular, $S^ax,\ S^bx \notin \partial M$ and its trajectory segment
is rich. Then there exist a neighbourhood $U \subset M$ of $x$,
and a submanifold $N$ such that
\roster
\item $\text{\rm codim}\ N \ge 2$;
\item for every $y \in U \setminus N$, \ \ $S^{[a,b]}y$ is sufficient.
\endroster
(As to the definition of sufficiency cf. Def. 2.4 of [KSSz-92].)
\endproclaim
The demonstration of Main Lemma 2.2 will be the content of section 3.
Denote by $M^0_p$ the subset of poor phase points from $M^0$. It would
be nice to claim that $M_p^0$ is residual but there
may exist some
trivial one-codimensional submanifolds of non-sufficient points for
our billiard (and for
some auxilliary sub-billiards used in the proof as well). Therefore we
should exclude a finite union of one-codimensional submanifolds
to obtain $M^\#$. Since the introduction of these submanifolds requires
some preparation, it is postponed to Appendix 2. Until then it may be
instructive to remark that this finite union necessarily contains all phase points whose trajectories never collide in at least one non-trivial sub-billiard
of our system. As such these submanifolds are defined
by linear conditions on the velocities.
Thus we can only claim
\proclaim {Main Lemma 2.3} $M^0_p \cap M^\#$ is a residual subset.
\endproclaim
\noindent whose demonstration can be found in Section 4.
The reader familiar with the technique of establishing global ergodicity
of semi-dispersing billiards already knows that the treatment
corresponding to the previous main lemmas for singular points on one
hand, and the verification of the Chernov-Sinai Ansatz, on the
other hand, follow from
\proclaim {Main Lemma 2.4} For every cell $C$ of maximal dimension
$2d-3$ in $\sr$, the set $C_{ed} \subset C$ of all eventually
disconnected phase points can be covered by a countable family of closed
zero-subsets (with respect to the surface measure $\mu_C$ in $C$) of
$C$.
\endproclaim
We say that a point $x \in \sr$ is {\it eventually disconnected} if
\roster
\item the semi-trajectory $S^{{\Bbb R}_+}x$ is regular;
\item there is a number $t_0>0$ such that the trajectory segment
$S^{(t_0,\infty)}x$ is not connected.
\endroster
As said earlier, Main Lemma 2.4 can be derived in almost exactly the
same way as Main Theorem 6.1 in [KSSz-92], and, for brevity, we are
satisfied by giving some indications at the end of section 4.
Given Main Lemmas 2.2-2.4, the proof of our Theorem is
straightforward.
It is, first of all, easy to see that the Transversal Fundamental
Theorem (for simplicity consider its form given in Theorem 3.4 of
[KSSz-92]) is applicable. Indeed, by Main Lemma 2.4, the
Chernov-Sinai Ansatz 3.1 holds
whereas the fulfillment of the geometric conditions
3.2-3.3 can be seen by standard arguments. Further, our Main Lemmas
2.2-2.4 are clearly
the analogues of Main Theorems 4.3, 5.1 and 6.1 of [KSSz-92].
Consequently, they provide the proof of the fact that every connected
component $\Omega_1, \dots, \Omega_I$ of the set $M^\#$ introduced in
Appendix 2 belongs to one ergodic component. Indeed, this results in
the same way
as, in section 7 of [KSSz-92], the aforementioned Main Theorems lead to
the proof of the Main Theorem of that work.
These components, however, can be connected by bundles of orbits of
positive measure as also explained in Appendix 2. This remark already
gives the statement of our Theorem. \hfill \qed
At this point it is time to mention how the assumption on the
disjointness of the cylinders can be weakened. If we drop it, then,
in principle, there may be trajectories with an infinite number of
collisions in a finite time interval. Therefore, as usual, we put,
first of all, the
following condition (Condition 2.1 from [KSSz-90]):
{\it The set of phase points in whose trajectories the moments
of reflections accumulate in a finite time interval is a residual
subset.}
Under this condition the proof of our Theorem automatically provides that, in
general, the number of ergodic components is finite.
Furthermore, the method described in
Appendix 2 for connecting different ergodic components definitely works if we,
in addition, require e. g. that
$$
\{K^j:\ \ C^j \ \ \text {\rm does not intersect any other cylinder} \}
$$
is itself a connected cover of $\{1, \dots , d\}$.
\vskip1cm
\heading 3. Geometric-algebraic considerations: Proof of Main Lemma 2.2
\endheading
\medskip
Throughout the whole section we will only consider regular trajectory
segments. Main Lemma 2.2 will follow from the following
\proclaim {Lemma 3.1} Assume that the trajectory segment
$S^{[a,b]}x$ is
a regular, connected one with
$S^ax,\ S^bx \notin \partial M$. Then there exists a neighbourhood $U \subset M$ of $x$,
and a submanifold $N$ such that
\roster
\item $\text{\rm codim}\ N \ge 1$;
\item for every $y \in U \setminus N$, $S^{[a,b]}y$ is sufficient.
\endroster
\endproclaim
{\it Proof of Lemma 3.1.} We start by characterizing the neutral
subspaces of trajectory segments with simple collision sequences. Assume
first that $S^{[a,b]}x,\ S^ax,S^bx \notin \partial M$ contains a single
collision at time $\tau \in (a,b)$ with the cylinder
$C:=\sigma_u(B \times A)$ where $B:= \{(q_i: i \in K):\ \ (\sum_{i
\in
K} q^2_i)^{1/2} \le r\}$ and $A:=\{ (q_i: i \in K^c)\}$. Now
from the definition of the neutral subspace $W_t(S^{[a,b]}x)$
(cf. Def. 2.2 in [KSSz-92])
the following statement is obvious:
\proclaim {Sublemma 3.2} If the trajectory segment
$S^{[a,b]}x,\ S^ax,S^bx \notin \partial M$ contains a single collison
as above, then
$$
W_{\tau^\pm}(S^{[a,b]}x)= \{\alpha P_K(V^\pm_{\tau}) + P_{K^c}(Z):\ \
\ \alpha \in \r, Z \in \r^d\} \tag{3.3}
$$
where, in general, $P_K: \r^d \to \r^d$ denotes the orthogonal
projection to the subspace $\{Q=(q_1, \dots, q_d):\ \ q_i=0\ \
\text{if}\ \ i \in K^{c}\}$, and $V^{\pm}=p(S^{\tau\pm}x)$.
\endproclaim
{\it Remark 3.4.} Clearly, for any $W \in W_{\tau^{\pm}}(S^{[a,b]}x)$,
the value of
$\alpha:=\alpha(W)$ is uniquely determined and $\alpha:\
W_{\tau^{\pm}}(S^{[a,b]}x)
\to \r$ is a linear functional. Indeed, the uniqueness of $\alpha$ is
evident from (3.3) whenever $P_K(V)\neq 0$. But $P_K(V) = 0$ is
impossible
since then $V^{\pm}_\tau\|A$ and no collison occurs with the cylinder
$C$.
This functional is related to the notion of the
{\it advance} of a collision introduced, in general, for any collision
of an
arbitrary trajectory segment in [KSSz-92] and in [S-92]. It is easy to see
that, for the particular case of
Lemma 3.1, $\alpha(W)={ {(P_K(W),V)} \over {\|P_K(V)\|^2}}$.
Next assume that, for some $a < t < b$, both trajectory segments
$S^{[a,t]}x$ and
$S^{[t,b]}x$ with
$(S^ax, S^tx, S^{b}x
\notin \partial M)$ contain exactly one collision in the times
$a < \tau < t < \tau' < b$ with the cylinders
$C:=\sigma_u(B \times A)$ and $C':=\sigma_{u'}(B'
\times A')$, respectively. Now, if $W \in W_t(S^{[a,b]}x)$, then for
some $\alpha, \beta \in \r;\ \ Z, Z' \in \r^d$
$$
W = \alpha P_K(V_t) + P_{K^c}(Z) = \beta P_{K'}(V_t) +
P_{(K')^c}(Z').
$$
This equation then leads to
$$
(\alpha - \beta) P_{K \cap K'}(V_t)
+ P_{K \setminus K'}(\alpha V_t - Z')
+ P_{K' \setminus K}(- \beta V_t + Z)
+ P_{(K \cup K')^c}(Z - Z') = 0.
$$
>From this relation one readily obtains
\proclaim {Sublemma 3.5} If the trajectory segment $S^{[a,b]}x$
contains exactly two collisions as above, and $P_{K \cap K'}(V_t) \neq
0$, then $\alpha = \beta$ and
$$
W_t = \alpha P_{K \cup K'}(V_t) + P_{(K \cup K')^c}(Z)\tag{3.6}
$$
where $\alpha \in \r, Z \in \r^d$.
\endproclaim
A similar argument provides the validity of
\proclaim {Sublemma 3.7} Consider a trajectory segment $S^{[a,b]}x$
with a collision sequence $(C_1, \dots , C_k)$ ($C_i \in
\{C^j: 1 \le j \le J\}:\ \ 1 \le i \le k)$
and collision times $a < \tau_1 < \dots \tau_k < b$. Denote the
non-trivial connected components of the family
$\{K_i:\ \ 1 \le i \le k\}$
of subsets by $H_1, \dots , H_m: \vert H_i \vert \ge 2$ and let $H_0 :=
\{1,
\dots, d\}
\setminus \cup_1^mH_i$. Assume further that,
for any $1 \le i \le k-1$ and
any $1 \le l \le d$, we have $v_l^+(\tau_i) \neq 0$.
Then
$$
W_b(S^{[a,b]}x) = \{ \sum _{g=1}^m \alpha_g P_{H_g}(V_b) + P_{H_0}(Z):\ \
\alpha_g \in \r,\ \ Z \in \r^d\}. \tag{3.8}
$$
(A similar representation is true for $W_a(S^{[a,b]}x)$, too.)
\endproclaim
An important particular case of this sublemma is
\proclaim {Corollary 3.9} In the setup of Sublemma
3.7, assume further that
$\{K_i:\ \ 1 \le i \le k\}$ is a connected cover of $\{1, \dots, d\}$.
Then the trajectory segment is sufficient.
\endproclaim
Proof of Sublemma 3.7. We proceed by induction on $k$. The case
$k=0$ is trivial. Assume
that the statement is true for any $k' \le k$ and any interval $[a,b]$.
Let $a < \tau_1 < \dots < \tau_k < b < \tau_{k+1} < b'$. By our
inductive assumption, (3.8) holds true. If $ W \in W_b(S^{[a,b']}x) $
then, on one hand, it should be of the form (3.8) and, on the other
hand,
of the form $\beta P_{B_{k+1}}(V_b) + P_{A_{k+1}}(Z')$. The argument
of Sublemma 3.5 then gives $\alpha_g = \beta\ \ \text{whenever}\ \
H_g
\cap K_{k+1} \neq \emptyset$. We obtain, moreover, $W_b(S^{[a,b']}x)$ as
a sum analogous to (3.8), from where the required form of
$W_{b'}(S^{[a,b']}x)$
comes by reflection with respect to $C_{k+1}$. \hfill \qed
Corollary 3.9 easily provides now the {\it proof of Lemma 3.1}. Indeed,
we can assume $b=0$. In a small neighbourhood $U$ of $x$ the collision
sequence of $S^{[a,0]}y$ is a constant. For $(x \in) U$ sufficiently
small, let
$N:=\
\{y:\ \ v_l^+(\tau_i) = 0 \ \ \text{for some}\ \
1 \le i \le k-1\ \ \text{and some}\ \ 1 \le l \le d\}$.
By Sublemma 3.7,
of course, every $y \in U \setminus N$ is sufficient. On the other hand,
for any $ 1 \le i \le k-1$\ and any $ 1 \le l \le d$, \ \
$\text{codim} \{y:\ \ v_l^+(\tau_i) = 0\}$ is 1 since, in a small
neighbourhood, $S^{\tau_i}y$ is a diffeomorphism. Hence Lemma 3.1.
\hfill \qed
We can next turn to the
Proof of Main Lemma 2.2. Assume, for simplicity, that $a < 0 < b,\ x
\in \partial M^0,$
and the trajectory segments $S^{[a,0]}x$ and $S^{[0,b]}x$ are connected.
Denote their collision sequences and collision times by
$(C_{-k^-}, \dots , C_{-1})$, $(C_1, \dots , C_{k^+})$ and
$a < \tau_{-k^-} < \dots \tau_{-1} = 0 = \tau_1 < \dots \tau_{k^+}
< b$, respectively.
Introduce, moreover, the notations
$$
N^-:=\ \{y:\ \ v_l^-(\tau_i) = 0 \ \ \text{for some}\ \
-k^-+1 \le i \le -1\ \ \text{and some}\ \ l \in K_i\}
$$
$$
N^+:=\
\{y:\ \ v_l^+(\tau_i) = 0 \ \ \text{for some}\ \
1 \le i \le k-1\ \ \text{and some}\ \ l \in K_i\}
$$
and
$$
N:= N^- \cap N^+
$$
It is evident from the proof of Lemma 3.1, that, if $y \in U
\setminus N$, then $y$ is sufficient. It remains to show that
$\text{codim}\ N \ge 2$. This statement, however, will be a straight
consequence of
\proclaim {Lemma 3.10} Suppose that $S^{[0,b]}x^0$ is a regular
trajectory segment with a collision sequence $(C_1, \dots , C_k)$.
Assume further that either $l \neq 1$, or if $l = 1$, then
$ 1 \in \cup_{i=1}^k K_i$. Under these conditions there exists a
neighbourhood $U$ of $x^0$ such that
$$
\text{\rm codim}_U\ \{v_1(0)=0,\ \ v_l(b)= 0\} \ge 2 \tag {3.11}
$$
\endproclaim
Proof of Lemma 3.10. The argument is again inductive. We assume that the
claim
holds true in any dimension $2 \le d' < d$ \ \ for any number $k( \ge
0)$ of collisions, and also in dimension
$d$ if the number of collisions $k' < k$. Thus we can also suppose that
$\{K_i:\ \ 1 \le i \le k\}$ is a cover of $\{1, \dots, d\}$. If $1$ and
$l$ belong to different connected components of this cover, then the
statement of the lemma is obvious since then the dynamics is the product
of two or more dynamics and the subset in (3.10) is a product set of
two, one-codimensional submanifolds in the phase spaces of these
noninteracting dynamical systems. Consequently, we can and do
assume that $\{K_i:\ \ 1 \le i \le k\}$ is a connected cover of
$\{1, \dots, d\}$.
The non-trivial case is, of course, $v_1^0(0) = 0$.
Therefore we will consider a product type neighborhood $\bar U = \bar U_{Q^0}
\times \bar U_{V^0}$ in the one-codimensional submanifold determined by the
condition
$v_1(0) = 0$ of a point $x^0:=(Q^0,V^0)$ satisfying $v_1^0(0) = 0$. (
Here
$\bar U_{Q^0}$ and
$\bar U_{V^0}$ belong to the configuration and the velocity spaces,
respectively.)
Denote
$$
\bar N := \{y:\ v_j^+(\tau_i)=0\ \ \ \text {\rm for some}\ 1 \le i
\le k-1
\ \ \text {\rm and some}\ \
j \in K_i\}.
$$
By our inductive assumption and Lemma A.1.1
$$
\text {\rm codim}_{\bar U}\ \bar N\ = 1.
$$
Consequently, by Lemma A.1.2, there exists a $\bar Q \in \bar U_{Q^0}$
such that
$$
\text {\rm codim}_{\bar U_{V^0}}\ (\bar N)_{\bar Q}\ = 1.
$$
Now, by Corollary 3.9, if $V \in \bar U_{V^0} \setminus (\bar N)_{\bar Q}$, then
$S^{[0,b]}(\bar Q,V)$ is a sufficient trajectory segment. By sufficiency
then, for
these $V$, the map of the neighborhood of the origin of $\Bbb R^{d-1}$,
consisting of
vectors $\{dQ; (dQ,V)= 0\}$, into $S_{d-1}$ defined by
$$
p(S^b(\bar Q + dQ,V))
$$
is a local diffeomorphism (following the traditions, $p:\ M \rightarrow
S_{d-1}$ is defined by $p(Q,V):= V$).
Thus the inverse image of the one-codimensional
submanifold of $S_{d-1}$ defined by the equation $v_l(b) = 0$ is also
one-codimensional for every $V \in \bar U \setminus (\bar N)_{\bar Q}$.
Finally, by Lemma A.1.3, we obtain that
$$
\text {\rm codim}_{\bar U}\ \{y:\ p_l(S^by) = 0\} = 1
$$
and thus, by Lemma A.1.4, we obtain the desired statement.
\hfill \qed
\vskip1cm
\heading 4. Dynamical-topological part: Proof of Main Lemma 2.3
\endheading
We need first a simple characterization of poor trajectory segments.
\proclaim {Lemma 4.1} If a regular orbit $S^{\Bbb R}x$ is poor, then there
exists
a non-collision moment $\tau$ such that neither of the trajectory segments
$S^{(-\infty,\tau)}x$ and $S^{(\tau,\infty)}x$ is connected.
\endproclaim
Proof: Denote
$$\alignat 4
T^-&:=\ &&\text {\rm sup}&&\{t:\ S^{(-\infty,t)}x\ &&\text {\rm is not
connected}\}\\
T^+&:=\ &&\text {\rm inf}&&\{t:\ S^{(t,\infty)}x\ &&\text {\rm is not
connected}\}\endalignat
$$
The statement is trivial if any of $T^-$ or $T^+$ is not finite.
If $T^-$ and $T^+$ is finite, then it is necessarily a collision time. It is
easy to see that the orbit $S^{\Bbb R}x$ is poor if and only if $T^- >
T^+$, thus providing the statement of the lemma.
\hfill \qed
\medskip
For a poor regular orbit, the value of $\tau$ figuring in the statement of
Lemma 4.1 can be chosen to be rational. By Property 2.9 of
[KSSz-91], it is sufficient to show the validity of
\proclaim {Lemma 4.2}
$$
M_{nc}:= \ \{x\in M^0 \setminus \partial M:\ S^{(-\infty,0)}x\ \ \text
{\rm and}\
\ S^{(0,\infty)}x\ \text {\rm are not connected}\}
$$
is a residual subset.
\endproclaim
Proof of Lemma 4.2. Let $x \in M$. We will repeatedly use Lemma 2.14 of
[KSSz-91] ensuring that residuality be checked locally.
Denote the partition into connected components of the collision graphs
of $S^{(-\infty,0)}x$ and of $S^{(0,\infty)}x$ by
$$
P^-:=\ \{H^-_1, \dots, H^-_{i^-}\}\ \ \text {\rm and by }\ \ P^+:=\
\{H^+_1,
\dots, H^+_{i^+}\}\tag{4.3}
$$
respectively. We wish to separate the cases
\roster
\item "{(i)}" $P^- \wedge P^+ = \epsilon$ , where $\epsilon$ denotes the
trivial partition of $\{1, \dots, d\}$, and
\item "{(ii)}" $P^- \wedge P^+ \neq \epsilon$
\endroster
\medskip
Let us start with the simpler case (ii).
Here we can suppose that there
exist nonempty $H_1$ and $H_2$ with $\ H_1 \cap H_2 = \emptyset,\ \ H_1 \cup H_2 =
\{1, \dots, d\}$ such that they are both $P^--$ and $P^+-$ measurable.
Under this condition, a variation of the ball-avoiding argument leads
us to our goal.
Indeed, under the assumptions of the
theorem, there exists a $j_0$ such that $K^{j_0}$ intersects both $H_1$
and $H_2$. Since we are considering phase points where the
dynamics
$$
S^ty =(S^t_{H_1}y^{}_{H_1}, S^t_{H_2}y^{}_{H_2})\tag{4.4}
$$
is the product of two non-interacting evolutions, the quantities
$$
\sum_{i \in H_l} v_i^2 = E_l, \qquad \ \ l=1,2 \tag {4.5}
$$
are conserved. Fix the values of
$E_1$ and $E_2$\ $(E_1+E_2=1)$ and restrict our considerations to the
submanifold $\Phi:= M_{H_1,\ E_1,\ H_2,\ E_2}$ of $M$ where (4.5) holds.
The phase points of the aforementioned evolutions
necessarily the open subset
$$
B:= \{x \in \Phi:\ \text {\rm for every}\ t\in {\Bbb R}\ \ \ \
(\sum_{l \in K^{j_0}}\ (q^t_l - u^{j_0}_l)^2)^{1/2} < r^{j_0}\}.
$$
By the inductive assumption the systems $M_{H_i,E_i},S^\Bbb R_{H_i},
\mu_{H_i}),\ \ i=1,2$ are K (here we assumed $\Vert H_i \Vert \ge 2$; if
$\Vert H_i \Vert =1$, then the adaptation of the ideas of [KSSz-89] to
our situation is even simpler). Their product (4.4) is consequently
mixing and, moreover, as it is easy to see, it is a hyperbolic system
with singularitites. More exactly, $\text {\rm dim} \Phi = 2d-2$ and for
the product $\dim \gamma^s,u=d-2$ whereas $\dim \gamma^0=2$. Now we
claim that
\proclaim {Lemma 4.6} Consider the product of two ergodic
semi-dispersing billiards, each satisfying the conditions of the
fundamental theorem (Theorem 3.4 from [KSSz-92]). Let $B$ be an open set
in the phase space of the product system and $H \subset \Bbb R$ such
that $\inf H = -\infty, \ \ \sup H = \infty$ and, moreover
$$
A_H(B):=\ \ \{x \in \Phi:\ \ S^Hx \cap B = \emptyset\}.
$$
Then $\text {\rm codim}_\Phi\ A_H(B) \ge 2.$
\endproclaim
The proof of the lemma is completely analogous to that of Lemma 3 of
[KSSz-89]. The only difference is that here we have two neutral
directions corresponding to the time advances of both factor-dynamics.
The natural way out is to consider sections in both factors.
By choosing $H=\Bbb R$ and applying Lemma 4.6 and then Lemma A.1.7 we
obtain the statement of Lemma 4.2 in case (ii).
\medskip
Turn now to case (i). First we formulate an important lemma. Denote
$$
F:= \{y:\ \ S^{{\Bbb R}_\mp}y\ \ \text {\rm is partitioned by} \ \
P^{\mp}\}
$$
\proclaim {Lemma 4.7} Suppose that $x \in M$ and the intervals $I^-:=
[a,0]$ and
$I^+:= [0,b], \ \ (a<0** 0$. On the other
hand, (4.13) trivially implies $(dQ,dV^-) = 0$, a contradiction.
Hence Sublemma 4.11. \hfill \qed
\medskip
To complete the proof of Lemma 4.7, we note that if either $i^-$ or $
i^+
\neq
2$, then $\gamma^s$ and $\gamma^u$ can be defined by a natural
generalization of (4.8) and their maximal dimensions are
$$
\text {\rm dim} \gamma^u = \sum_{i=1}^{i^-} (d^-_i - 1) = d - i^-
$$
$$
\text {\rm dim} \gamma^s = \sum_{i=1}^{i^+} (d^+_i - 1) = d - i^+
$$
Moreover, the generalization of the arguments around (4.10) easily
provides that $\text {\rm dim} \gamma^0 = i^- + i^+ -1.$ Lemma 4.11
holds true in this general case, too, and, since $(d-i^-) +(d-i^+) +
(i^- + i^+ -1) = 2d - 1$, the methods of [KSSz-92] are again applicable.
Hence Lemma 4.7. \hfill \qed
\medskip
Our final task is to argue that Lemma 4.7 implies the validity of Lemma
4.2 in case (i). To that end we recall again Lemma 2.14 of [KSSz-91] saying
that residuality of a set $A$ can be checked locally, i. e. in
neighborhoods of points $x \in A$. Fix a point $x \in M_{nc}$. There
necessarily exist the limit partitions
$$
P^{-\infty}:=\ \ \lim_{t \to -\infty}\ \ P(S^{(-\infty, t)}x)
$$
and
$$
P^{\infty}:=\ \ \lim_{t \to \infty}\ \ P(S^{(t,\infty)}x)
$$
where $P(S^{[a,b]}x)$ denotes the partition into connected components of
the collision graph of $S^{[a,b]}x$. In general,
$P^-\preceq P^{-\infty}$ (and $P^+\preceq P^{\infty})$. By definition,
there exist finite rational intervals
$I^-:= [t^-_1, t^-_2] \subset {\Bbb R}_-$ and
$I^+:= [t^-_1, t^-_2] \subset {\Bbb R}_+$ such that $P(S^{I^\mp}x) =
P^{\mp \infty}$ and, moreover, the collision graph of each
$S^{[t^-_1-t^-_2,0]}_{H^{-\infty}_i}(S^{t^-_2}x)_{H^{-\infty}_i},\ \
1\le i \le i^-$
and of
$S^{[0,t^+_2-t^+_1]}_{H^{\infty}_l}(S^{t^+_1}x)_{H^{\infty}_l},\ \
1\le l \le i^+$
is rich with respect to the corresponding subdynamics. In virtue of Main
Lemma 2.2, by discarding a suitable residual subset, we can assume that
all aforementioned trajectory segments are sufficient in the
corresponding subdynamics.
In case (i), if $S^{[t^-_2,t^+_1]}x$ does not contain any collision,
then, by Lemma 4.7, we are done. If it contains, then let us assume that
these
collisions occur at times $(t^-_2 <) \tau_1 < \dots \tau_m (< t^+_1)$
with
the cylinders $C_1, \dots, C_m$. For all $1 \le i \le m$ consider the
velocities $v_l(\tau_i+):\ \ l \in K_i$ and, moreover, the velocities
$v_l(t^-_2+):\ \ 1 \le l \le d$. By virtue of Lemma 2.10, apart from a
residual subset, at most one of all these velocities vanishes. Assume
that it is $v_l(\tau_i+)$ for some $1 \le i \le m,\ l\in K_i$, say.
Then choose $\tau_i < \bar t <\tau_{i+1}$, and consider the trajectory
segment $S^{[t^-_1,t^+_2]}x$. Now Lemma 4.7 can be applied to the
trajectory segments $S^{[t^-_1,\bar t]}x$ and $S^{[\bar t, t^+_2]}x$
since, by Lemma 3.7, all the sub-billiard trajectories of these
trajectory segments are sufficient (the sub-billiard trajectories
should, of course, be taken on connected components of the collision
graphs of $S^{[t^-_1,\bar t]}x$ and of $S^{[\bar t, t^+_2]}x$).
This completes the discussion of case (i). Hence Lemma 4.2.
\hfill \qed
\medskip
{\bf Singular orbits.}
To end this section we turn to some thoughts about the proof of Main
Lemma 2.4. Our subset $C_{ed}$ of eventually disconnected points is
analogous to the subset $C_{ed}$ of eventually decomposing points used
in [KSSz-92]. For any $t_0 \in {\r}$ and any partition $P$ of
$\{1, \dots, d\}$ we can also introduce the subset
$$ \aligned
C_{ed}(t_0,P) := \ \ \{x \in C_{ed}:\ \ & \text {\rm the partition into
the connected }\\ &
\text {\rm components
of the collision graph of} \\ & S^{(t_0,\infty)}x \ \
\text
{\rm is finer than}\ \ P\} \endaligned
$$
Assume, for simplicity, that $P=\{H_1,H_2\}$ has two nontrivial
elements. Now the definitions of $U(y_0), F_+, F'_+$ are evidently the
same as in [KSSz-92], whereas $\gamma^s_0(.)$ is given by (4.5), and
further $\gamma_{H_i}^s(.):i = 1,2$ will appear instead of the manifolds
$\gamma_{1,2}^s(.)$ and $\gamma_{3,4}^s(.)$. The dimension of
$$
\gamma^{sp}_0(y):= \{z \in \gamma^s_0(y): Q(z) -Q(y) \perp V(y)\}
$$
is, of course, 1 and the dimension of
$$
\g ^s_e (y): =\bigcup_{z\in \g ^s_{H_1}(y)} \g^s_{H_2}(z)\,, \tag {4.13}
$$ is $d-2$. That of the generate
$$
\g ^s_g (y):= \bigcup\limits_{z\in \g^{sp}_0 (y)} \g^s_e(z)=
\bigcup\limits_{z\in \g^s_e(y)} \g^{sp}_0(z)
$$
is then $d-1$. It is easy to see that, with these modifications in the
definitions, the proof of Main Theorem 6.1 can literally be copied.
\vskip1cm
{\bf Acknowledgements.} The author is indebted to John Mather
for raising the question which was actually the starting point of this
research. Sincere thanks are due to N\'andor Sim\'anyi for his most critical
reading of the first version of the text and for his remark that the
analogue of the Theorem of the Appendix of [KSSz-92] fails to hold for
cylindric billiards. Useful remarks of the referee are also
acknowledged. Finally, the author expresses his gratitude for the
hospitality
of the IHES, Bures-sur-Yvette where, on one hand, Section 3 was written during
the author's stay in November 1992 and, on the other hand, the final version
was prepared in April 1993.
\bigskip
\heading References.
\endheading
\medskip
\ref\key BLPS-92
\by \quad\quad\quad\quad\quad L. Bunimovich, C. Liverani, A. Pellegrinotti, Yu. Sukhov
\paper Special systems of hard balls that are ergodic
\jour Commun. Math. Phys.
\vol 146
\pages 357-396 (1992)
\endref
\ref\key BS-73
\by \quad\quad\quad\quad\quad L. A. Bunimovich, Ya. G. Sinai
\paper On a Fundamental Theorem in the Theory of Dispersing Billiards
\jour Mat. Sbornik
\vol 90
\pages 415--431 (1973)
\endref
\ref\key E-78
\by \quad\quad\quad\quad\quad R. Engelking
\paper Dimension Theory
\jour North Holland,
\pages 1978
\endref
\ref\key KSSz-89
\by \quad\quad\quad\quad\quad A. Kr\'amli, N. Sim\'anyi, D. Sz\'asz
\paper Ergodic Properties of Semi--Dispersing Billiards I.
Two Cylindric Scatterers in the 3--D Torus
\jour Nonlinearity
\vol 2
\pages 311--326 (1989)
\endref
\ref\key KSSz-90
\by \quad\quad\quad\quad\quad A. Kr\'amli, N. Sim\'anyi, D. Sz\'asz
\paper A ``Transversal" Fundamental Theorem for Semi--Dispersing Billiards
\jour Commun. Math. Phys.
\vol 129
\pages 535--560 (1990)
\endref
\ref\key KSSz-91
\by \quad\quad\quad\quad\quad A. Kr\'amli, N. Sim\'anyi, D. Sz\'asz
\paper The K--Property of Three Billiard Balls
\jour Annals of Mathematics
\vol 133
\pages 37--72 (1991)
\endref
\ref\key KSSz-92
\by \quad\quad\quad\quad\quad A. Kr\'amli, N. Sim\'anyi, D. Sz\'asz
\paper The K--Property of Four Billiard Balls
\jour Commun. Math. Phys.
\vol 144
\pages 107-148 (1992)
\endref
\ref\key M-91
\by \quad\quad\quad\quad\quad J. Mather
\jour Oral communication
\endref
\ref\key S-70
\by \quad\quad\quad\quad\quad Ya. G. Sinai
\paper Dynamical Systems with Elastic Reflections
\jour Usp. Mat. Nauk
\vol 25
\pages 141--192 (1970)
\endref
\ref\key S-92
\by \quad\quad\quad\quad\quad N. Sim\'anyi
\paper The K-property of $N$ billiard balls I
\jour Invent. Math.
\vol 108
\pages 521-548 (1992)
\moreref
\paper II.
\jour ibidem
\vol 110
\pages 151-172 (1992)
\endref
\ref\key SCh-87
\by \quad\quad\quad\quad\quad Ya. G. Sinai, N.I. Chernov
\paper Ergodic Properties of Some Systems of 2--D Discs and 3--D Spheres
\jour Usp. Mat. Nauk
\vol 42
\pages 153--174 (1987)
\endref
\ref\key Sz-93
\by \quad\quad\quad\quad\quad D. Sz\'asz
\paper Ergodicity of Classical Hard Balls
\jour Physica {\bf A}
\vol 194
\pages 86-92 (1993)
\endref
\bigskip
\heading Appendix 1. Some facts from topological dimension theory.
\endheading
\medskip
In the proofs some additional facts are needed compared to those discussed in
[KSSz-92]. Here only these will be treated otherwise the reader is
suggested
to consult the aforementioned paper or the monograhpy [E-78].
Moreover, for simplicity
of exposition, we do not aim at the widest possible generality. In particular,
we always assume that our basic space $M$ is a compact smooth manifold
and, moreover, it is the product $M = M_1 \times M_2$ of smooth manifolds.
Denote, for any $A \subset M$
$$
\text {\rm codim}_M\ A := \text {\rm dim}\ M - \text {\rm dim}\ A
$$
where $\text {\rm dim}\ A$ is any of the three classical notions of dimension:
the covering,
the small inductive or the large inductive dimension.
\proclaim {Lemma A.1.1} Assume that
\roster
\item "{(i)}" $F$ is a subset of $M$ with $\text {\rm codim}_M\
F
\ge 2$;
\item "{(ii)}" $\bar M$ is a submanifold in $M$ with $\text {\rm
codim}_M\ \bar M = 1$.
\endroster
Then $\text {\rm codim}_{\bar M}\ F \ge 1.$
\endproclaim
\proclaim {Lemma A.1.2} Assume $F$ is a closed subset of $M = M_1 \times M_2$
with \break
$\text {\rm codim}_M\ F \ge 1$. Then there exists an $x \in M_1$ such that
$\text {\rm codim}_{M_2}\ F_x \ge 1$, where $F_x := \{y:\ (x,y) \in
F\}.$
\endproclaim
\proclaim {Lemma A.1.3} Assume that $A$ is an arbitrary subset of $M = M_1
\times M_2$,
and
\roster
\item "{(i)}" there exists an $L \subset M_2$ satisfying $\text {\rm
codim}_
{M_2}\ L \ge 1$;
\item "{(ii)}" for every $y \in M_2 \setminus L$ we have $\text {\rm
codim}_
{M_1}\ A_y \ge 1$,
where $A_y := \{x:\ (x,y) \in A\}$.
\endroster
Then $\text {\rm codim}_{M}\ A \ge 1$.
\endproclaim
\proclaim {Lemma A.1.4} Assume
\roster
\item "{(i)}" $\bar U$ is a smooth submanifold in $M$ with $\text {\rm
codim}_
{M}\ \bar U = 1$;
\item "{(ii)}" $F \subset \bar U$ is a subset such that $\text {\rm
codim}_
{\bar U}\ F \ge 1$
\endroster
Then $\text {\rm codim}_{M}\ F \ge 2$.
\endproclaim
These lemmas are simple consequences of the following characterizations of
one- and two-codimensional subsets.
\proclaim {Property A.1.5} For every subset $A \subset M$, $\text {\rm
codim}_
{M}\ A \ge 1$
if and only if \ \ $\text {\rm Int}\ A = \emptyset$.
\endproclaim
\proclaim {Property A.1.6} For any closed subset $F \subset M$, the following
three conditions are equivalent:
\roster
\item "{(i)}" $\text {\rm codim}_{M}\ F \ge 2$;
\item "{(ii)}" $F \neq M$ and, for every open connected set $G \subset
M$, the difference set $G \setminus F$ is also connected;
\item "{(iii)}" For every point $x \in M$ and for any neighborhood $V$
of $x$ in $M$
there exists a smaller neighborhood $W \subset V$ of the point $x$ such that,
for every
pair of points $y, z \in W \setminus F$, there is a continuous curve $\gamma$
in the set $V \setminus F$ connecting the points $y$ and $z$ and, moreover,
$\text {\rm Int } F = \emptyset$.
\endroster
\endproclaim
Indeed, Lemmas A.1.1 and A.1.4 are trivial, Lemma A.1.2 is a particular
case of Problem 1.9.C
of [E-78] and finally for Lemma A.1.3 we can argue as follows:
If $\text {\rm codim}_{M}\ A = 0$, then, by Property A.1.5, $A$ necessarily
contains a
subset $B_1 \times B_2$ where $B_1 \subset M_1,\ B_2 \subset M_2$ are
ball-like subsets.
If $\text {\rm codim}_{M_2}\ L \ge 1$, then, by the same property,
$(M_2 \setminus L) \cap B_2 \neq \emptyset$. If $y \in (M_2 \setminus L)
\cap B_2$,
then $\text {\rm codim}_{M_1}\ A_y \ge 1$, resulting in $\text {\rm Int} A_y =
\emptyset$ contradicting to the fact that for $y \in B_2\ \ B_1 \subset A_y$.
Finally, we formulate an additional useful property of codimension-two
sets from [KSSz-89] (Property 4 from section 4):
\proclaim {Lemma A.1.7} Assume $F$ is a closed subset of $M = M_1
\times
M_2$, and for every $y\in M_2$ \ \ $ \text {\rm codim}_{M_1}\ F_y \ge
2$. Then $ \text {\rm codim}_M\ F \ge 2$.
\endproclaim
\vskip1cm
\heading Appendix 2. Trivial non-sufficient points of the natural
sub-billiards.
\endheading
\medskip
For a nonempty subset $Z \subset \{1, \dots , J\}$ consider the
sub-billiard $(M_Z, S_Z^{\bold {R}}, d \mu_Z)$ on the
$d$-torus $\bold {T}^d$ with the cylinders $\{C^j:\ \ j \in Z\}$ as
scatterers.
\proclaim {Lemma A.2.1} The subset $\tilde {M}_Z$ of phase points from
$M_Z$ whose orbits have no collisions at all is contained in
a finite union $\tilde
{M}_Z:=\ \cup_{l=1}^{l_Z}\ E_l$ of submanifolds such that for every $l:\
1 \le l \le l_Z$
\roster \item
$$
\text {\rm codim} _{M_Z}\ E_l \ge 1;
$$
\item
$E_l^{cl}$ is compact.
\endroster
\endproclaim
The lemma is a direct consequence of
\proclaim {Lemma A.2.2} Let $D \subset \bold {T}^d$ be
any set with a nonempty interior. There exists a finite set of (unit) normal
vectors in ${\bold{R}}^d$ such that if the trajectory
$S^{\bold{R}}x$ of a phase point
$x=(Q,V)$
never enters $D$, then $V$ is orthogonal to at least one of the given normal
vectors.
\endproclaim
Proof of Lemma A.2.2. (joint with N. Sim\'anyi)
Introduce the natural coordinates in $\bold {T}^d$. We can assume that $D$ is
a small ball and $Q = 0$. The closure
$G=\{{\bold{R}}V\}^{cl}:= \{z \in {\bold{T}}^d:\ \
\text {\rm for some }\ \ t \in {\bold{R}} \ \
z = tV\}^{cl}$ of the one-parameter subgroup generated by $V$ is a subtorus
(connected closed subgroup) of $\bold{T}^d$. Furthermore it coincides with
${\bold{T}}^d$ whenever the coordinates of $V$ are
independent over the rationals. Suppose that this is not the case. Consider the
dynamical system $(\bold{T}^d, \Gamma^{\bold{R}}, dx)$
where $\Gamma_tx:= x + tV \ \ (\text {\rm mod }\ {Z}^d)$.
We want to find the invariant --- say $L_2$ --- functions
of $\tilde {\Gamma}_t : \tilde {\Gamma}_tf(x):= f(\Gamma _tx)$ by standard
Fourier methods. To that end we expand
$f(x)=\sum _{n} a_n exp(2\pi i (n,x))$. If $f$ is $\tilde {\Gamma}^{\bold{R}}$-
invariant, then $(n,V) \neq 0$ implies $a_n = 0$. Now $L:= \{n:\ (n,V) = 0\}
\subset {\bold{Z}}^d$ is a nontrivial subgroup and, in fact, by Pontryagin's
duality, its dual $L^*$ is just the factor group $\bold{T}^d/G$. This duality
also asserts that the subtorus $G$ is exactly the set of all points
$x\in \bold{T}^d$ for which all characters $\exp(2\pi i(n,x))$ with
$n\in L$ take on the value $1$. Consider now the natural projection
$\pi :\; \bold{R}^d\to \bold{T}^d$. The above characterization of $G$ yields
$$
\pi^{-1}(G)=\left\{x\in\bold{R}^d: \; (n,x) \text{ is an integer for every }
n\in L\right\}. \tag 1
$$
Set the orthogonal direct sum $\bold{R}^d=W\bigoplus W^{\bot}$, where the
subspace $W$ is the closure of all rational multiples of points from $L$.
Of course dim$W^{\bot}=$dim$G$. We denote the dimension of $W$ (i.e. the
dimension of $\bold{T}^d/G$) by $k$. It is clear that the group $L$ is
a lattice (a discrete subgroup of maximum possible rank $k$) in the
Euclidean space $W$. The characterization (1) of $\pi^{-1}(G)$ says
precisely that $\pi^{-1}(G)=L'\bigoplus W^{\bot}$, where the lattice
$L'$ in $W$ is the so called dual lattice of $L$, that is
$$
L'=\left\{x\in W: \; (n,x) \text{ is an integer for every } n\in L\right\}.
\tag 2
$$
Let us denote by $B$ one of the connected components of the inverse image
$\pi^{-1}(D)$. Thus, by the assumptions, the set $B$ is a ball of radius
$r$ which is disjoint from the set $\pi^{-1}(G)=L'\bigoplus W^{\bot}$.
Let us focus on the orthogonal projection $B_1=P_1(B)$ of the ball $B$
into the subspace $W$. By the disjointness just mentioned, we have that
in the k-dimensional Euclidean space $W$ there is a ball $B_1$ of radius
$r$ which is disjoint from the lattice $L'$ dual to the lattice $L$.
We claim now that there exists a finite number $f(k,r)$ (depending only on the
dimension of the Euclidean space and the radius of the ball) such that,
under the mentioned circumstances, one can find a nonzero vector
$n\in L$ for which $||n||\le f(k,r)$. By the appropriate scaling property
of the lattices (i. e. if we multiply $L$ by a positive number $\alpha$,
the corresponding dual lattice $L'$ will be multiplied by the reciprocal
of $\alpha$) we can easily see that {\bf the sharpest upper estimate}
$f(k,r)$ has the form $a_k/r$, where
$$
a_k=\sup \big\{||n||: \; n \text{ is the shortest nonzero vector of a
lattice } L
$$
$$
\text{ whose dual lattice does not intersect some ball of
radius one }\}.
$$
The value $a_1$ is, of course, $\dfrac{1}{2}$. The questioned finiteness
of the numbers $a_k$ is positively answered by
\proclaim{Lemma A.2.3} For every integer $k>1$
$$
a_k\le \sqrt{\frac{1}{4}+\frac{4}{3}a^2_{k-1}}.
$$
\endproclaim
\subheading{Proof:}
Suppose the finiteness of the right-hand-side.
Let $L$ be a lattice in the k-dimensional Euclidean space $W$ in which
the shortest nonzero vector $n_0$ is longer than
$$
\sqrt{\frac{1}{4}+\frac{4}{3}a^2_{k-1}},
$$
and let $B$ a (say open) ball of unit radius in $W$. We want to show that
the dual lattice $L'$ intersects $B$. Consider the set
$$
\left\{x\in W: \; (n_0,x) \text{ is an integer }\right\}. \tag 3
$$
This set is obviously a countable collection of equidistantly placed
hyperplanes in $W$, the distance between neighboring such hyperplanes
being $1/||n_0||$. Let $H$ be the one among these hyperplanes that has the
shortest distance from the center of the ball $B$, and let $H_0$ be the
other amongst these hyperplanes that contains the origin. Let, moreoverv,
$L'_0$ be the intersection of the dual lattice $L'$ with $H_0$. It is
clear that $L'_0$ is a lattice (of rank $k-1$) in $H_0$ and the intersection
$L'\cap H$ is a translated copy of $L'_0$. We denote the orthogonal projection
of the original lattice $L$ on the subspace $H_0$ by $L_0$. It is
straightforward that $L_0$ is a lattice in $H_0$ and its dual lattice in
the Euclidean space $H_0$ is just what we denoted by $L'_0$. Since every
point in the projection $L_0$ of $L$ is the projection of an appropriate
point of $L$ having distance from $H_0$ not bigger than $\dfrac{||n_0||}{2}$,
by the Pythagorean Theorem we get that the shortest nonzero vector of the
lattice $L_0$ can not be shorter than $\dfrac{\sqrt{3}}{2}||n_0||$, i.e.
its length is greater than
$$
\frac{\sqrt{3}}{2}\sqrt{\frac{1}{4}+\frac{4}{3}a^2_{k-1}}.
$$
By the definition of the number $a_{k-1}$, the lattice $L'_0$ must intersect
every open ball of radius
$$
\frac{a_{k-1}}{\frac{\sqrt{3}}{2}\sqrt{\frac{1}{4}+\frac{4}{3}a^2_{k-1}}}.
\tag 4
$$
However, by the choice of the hyperplane $H$ and by the Pythagorean Theorem,
the intersection of the unit ball $B$ with $H$ has radius not less than
$$
\sqrt{1-\frac{1}{4n_0^2}}
$$
which number is greater than
$$
\sqrt{1-\frac{1}{4(\frac{1}{4}+\frac{4}{3}a^2_{k-1})}}.
$$
Since this last number is exactly the right-hand-side of (4), the fact that
$L'\cap H$ is a translated copy of $L'_0$, immediately implies that
the lattice $L'$ intersects the ball $B$.
Hence the Lemma. \qed \hfill
Since $M, M_Z \subset \bold {T}^d \times S_{d-1}\ \ (Z \subset \{1,
\dots, J\})$, it makes sense to consider the connected components of the
set $$M^\#:=\ M \setminus \cup_{Z\neq \emptyset}\ (\tilde {M}_Z)^{cl}.$$
By Lemma A.2.1, their number is finite. Denote them by $\Omega_1, \dots,
\Omega_I\ \ (1 \le I < \infty)$.
The main body of the paper proves that each $\Omega_1, \dots, \Omega_I$
belongs to just one ergodic component. Here we add that they can be
connected by bundles of orbits of positive measure. This will be
clear from the elementary
\proclaim {Lemma A.2.3} Consider a billiard in $\bold {Q}:= \bold {T}^d
\setminus C$ where $C$ is a spherical scatterer. Vary the configuration
component $Q$ of a phase point $(Q, V^-)$ by keeping its velocity
coordinate fixed. Then, as $Q$ varies, $p(\tau^+(Q,V^-))$ is a mapping
with maximal rank $d-1$
(as usual, $p(\tau^+(Q,V^-))$ denotes the
outgoing
velocity after the first collision of the phase point $(Q,V^-)$).
\endproclaim
Indeed, each $\Omega_i:\ 1 \le i \le I$ is defined by a finite number of
linear inequalities on the velocities. On the other hand, by Lemma
A.2.3, for any $C^j:\ 1 \le j \le J$ the following statement is true:
for any fixed values of $v_i: i \notin K^j$ and for any pair $(v_i^-:\ i
\in K^j)$ and $(v_i^+: i \in K^j)$ such that for some pair $Q^-, Q^+$
$$
x^-:=\ (Q^-, \{v_i: \ i \notin K^j; v_i^-: i \in K^j \}) \in M^\#
$$
$$
x^+:=\ (Q^+, \{v_i: \ i \notin K^j; v_i^+: i \in K^j \}) \in M^\#
$$
where for some $\epsilon >0\ \ \ x^+ = S^\epsilon x^-$ and
$S^{0,\epsilon}x^-$ contains exactly one collison, it is true that
suitable neighbourhoods of the phase points $x^-$ and $x^+$ can be
connected by bundles of orbits of positive measure.
It is also clear that, if for some $j_1$ and $j_2$, $K^{j_1} \cap
K^{j_2}
\neq \emptyset$, then the aforeformulated statement also holds for the
union $K^{j_1} \cup K^{j_2}$ instead of $K^j$. Since $K^j: 1 \le j \le
J$ is a connected cover, we also obtain by induction that, in fact,
every connected component of $M^\#$ can be connected to any other by
bundles of trajectories of positive measure. Thus $M^\#$ makes, indeed,
just one ergodic component.
\end
**