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Cylindric Billiards, Hard Ball Systems, Ergodicity, Hyperbolicity
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\begin{document}
\title{Chaotic and Ergodic Properties of Cylindric Billiards}
\author{P\'eter B\'alint
\\ Alfr\'ed R\'enyi Institute of Mathematics, \\
Hungarian Academy of Sciences\\
H-1364, Budapest, P.O.B. 127, Hungary\\
E-mail: bp@math-inst.hu
}
\date{\today}
\maketitle
\begin{abstract}
Chaotic and ergodic properties are discussed in this paper for various
subclasses of cylindric billiards. Common feature of the studied systems is
that they satisfy a natural necessary condition for ergodicity and
hyperbolicity, the so called transitivity condition. Relation of our
discussion to former results on hard ball systems is twofold. On the one hand,
by slight adaptation of the proofs we may discuss hyperbolic and ergodic
properties of 3 or 4 particles with (possibly restricted) hard ball
interactions in any dimensions. On the other hand a key tool in our
investigations is a kind of connected path formula for cylindric billiards,
which, together with the conservation of momenta, gives back, when applied to
the special case of Hard Ball Systems, the classical Connected Path Formula.
\end{abstract}
{\it Keywords}: Cylindric Billiards, Hard Ball Systems, Ergodicity,
Hyperbolicity.
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\section{Introduction}
\label{secI}
\setcounter{equation}{0}
One of the most interesting open questions of statistical physics and
dynamical systems theory is the so-called Boltzmann-Sinai Ergodic hypothesis,
i.e. the conjecture that Hard Ball Systems in physical dimensions are
ergodic (as to the history of this conjecture see \cite{boltzmann}). In addition to
its physical relevance the question is interesting as it is highly nontrivial
from the mathematical point of view. Nevertheless, there is an increasing belief
in mathematical physics communities that the proof is within reach. Thus a
natural question is the following: what could be a suitable category of
dynamical systems containing all hard ball systems for which transparent
necessary and sufficient conditions for ergodicity can be proven?
Dynamical properties of hard ball systems are mainly characterized by the fact that
they belong to the category of semi-dispersive billiards. Nevertheless,
semi-dispersive billiards in their full generality may show a too extreme
variety of dynamical behaviour to guess the conditions for ergodicity. It has
turned out that one possible choice for the systems to answer the
question above is the class of {\it cylindric billiards} (see \cite{trans} or
conjecture~\ref{Ctr} in this paper).
The purpose of our study is to prove ergodicity for cylindric billiards with a
low number of scatterers (or, more precisely, with cylinders at most a low
number of which may have generator subspaces with nontrivial intersection, see Theorem~\ref{Te}). For some
systems (Theorem~\ref{Th}) instead of ergodicity only the weaker
hyperbolic/chaotic property is proven (from which by \cite{ks} it follows that
there are at most a countable number of ergodic components, on each of which
the dynamics possesses the K- and the B-properties). The most difficult steps of
the proofs (from the geometric-algebraic considerations) are discussed by the
help of a method for the calculation of neutral subspaces (Lemma~\ref{Lcpf})
which can be viewed as an analogue of the Connected Path Formula for Hard Ball
Systems (\cite{nandi}) in this cylindric billiard setting.
The paper is organized as follows. In Section~\ref{secAlt}
we summarize the most important prerequisites, state the results and make some
general remarks on the proofs. The proof of the above mentioned
Lemma~\ref{Lcpf} is also presented here (subsection~\ref{secAltalg}). In
Section~\ref{secErg} ergodicity and hyperbolicity is proven for cylindric
billiards with pairwise transversal generator subspaces
(Theorem~\ref{Te}). In Section~\ref{secHyp} we show hyperbolicity for
billiards with three cylinders (Theorem~\ref{Th}). The
sections consist of further subsections according to the steps of the
proofs. In the Appendix we summarize how our results are applicable to some
particle systems with hard ball interactions (together with some further simple
generalizations).
\section{Prerequisites and general observations}
\label{secAlt}
\setcounter{equation}{0}
\subsection{Definition of the dynamical system and summary of results}
\label{secAltre}
The subject of our study, the category of cylindric billiards is a simple
subclass of semi-dispersive billiards -- as to the notions in connection with
these systems see \cite{alapt}. In our case the configuration space of the
billiard is defined by cutting out a finite number of cylindric regions from
the $d$-dimensional unit torus, i.e. $Q={\bf T^d} \setminus (C_1
\cup\cdots\cup C_k)$. For the precise definition of the cylinders we need
three data for each $C_i$. We fix $A_i$, a subspace of the $d$-dimensional Euclidean
space ${\bf R^d}$, the so-called {\it generator subspace} of the
cylinder. $A_i$ should be a so-called
lattice-subspace to get a properly defined cylinder on
${\bf T^d}$ after factorization (\cite{trans}).
We assume $dim(L_i)\ge 2$, where $L_i=A_i^{\perp}$ is the notation for the {\it base subspace}, the orthogonal complement of $A_i$. The {\it base}, $B_i \subset L_i$ is a convex, compact domain, for
which, to ensure semi-dispersivity, the $C^2$-smooth boundary $\partial B_i$
is assumed to have everywhere positive definite second fundamental form. Furthermore a
translational vector $t_i \in {\bf R^d}$ is given to place our cylinder in
${\bf T^d}$. By the help of these data our cylinders are defined as:
$$
C_i:=\{a+l+t_i : a\in A_i, l\in B_i \} \big/ {\bf Z^d}
$$
To avoid possible complications we assume that:
(i) the domain $B_i$ does not contain any pair of points congruent modulo
${\bf Z^d}$; (ii) the interior of the configuration space $Q={\bf T^d} \setminus (C_1
\cup\cdots\cup C_k)$ is connected.
It is time to give the definition of our dynamical system $(M, S^{\bf R},
\mu)$. Our $2d-1$-dimensional phase space is the unit tangent bundle of $Q$,
i.e. $M=Q \times {\bf S^{d-1}}$ (here ${\bf S^{d-1}}$ is the $d-1$-dimensional
unit sphere). The dynamics $S^tx$ for a phase point $x\in M$ is understood in
continuous time and defined by uniform motion inside the domain and specular
reflections at the boundary (the cylinders). Finally, $\mu$ is, as usual, the
Liouville-measure (i.e. $d\mu = const \, dqdv$), which is invariant for the flow. For future convenience we
fix here some more notation. A finite trajectory segment, $S^{[a,b]}x$, is the
collection of points $S^tx$ on the trajectory of $x$ for which $a\le t\le
b$. For any phase point $x=(q,v)\in Q \times {\bf S^{d-1}}=M$ the natural
projections are defined as $p(x)=v$ and $\pi(x)=q$. By a non-trivial sub-billiard of our cylindric billiard we mean the billiard dynamical system we get by cutting out only some of the cylinders $C_i$.
\medskip
{\bf Example.} Consider $N$ ball particles with finite masses and radii moving
on the $\nu$-dimensional torus. To define dynamics, assume furthermore that a
so called {\it collision graph} (or graph of interactions), $\Gamma =({\cal
V},{\cal E})$ is given. Here ${\cal V}$ is the finite set of the $N$
particles. Pairs contained among the edges ${\cal E}$ do interact via hard
ball collisions, while non-connected pairs do not interact at all (see also
\cite{tollt}); otherwise the dynamics is governed by free motion. In the case
of a complete collision graph we get
the so deeply studied Hard Ball Systems. This dynamical system (restricted to the
constant value submanifold of the trivial integrals of motion) is equivalent to a
cylindric billiard with configuration space in ${\bf T^{N\nu-\nu}}$. The
cylinders we cut out correspond to the allowed interactions (in case the
configurational domain $Q$ is not connected -- i.e. the radii of the balls are
not small enough -- we may view the finitely many connected components of $Q$
as configurational domains for independent billiard systems). To get a detailed
description of this isomorphism see \cite{ngolyo} and references
therein.
Thus cylindric billiards are indeed generalizations of all possible Hamiltonian systems with restricted hard ball interactions. As to the relevance of our
results to this example see the Appendix.
\medskip
The two most important phenomena characterizing the dynamics in
semi-dispersive billiards is on the one hand that they enjoy some {\it hyperbolicity} (in fact, there exists an invariant, but not necessarily strictly invariant cone field, cf. \cite{lw}), and on the other hand that {\it
singularities} are present. As to the former one, among the most interesting questions of
the theory is whether hyperbolicity is strong enough to ensure that our
dynamical system is (i) completely {\bf hyperbolic}, i.e. with respect to
the invariant measure $\mu$ almost everywhere all relevant
Lyapunov-exponents of the flow are nonzero (such dynamical systems are sometimes referred to as chaotic); (ii) {\bf ergodic} with respect to the measure
$\mu$. There are two possible types of singularities in billiards. A collision
at the boundary point $(q,v)\in \partial M$ is said to be {\it multiple} if at
least two smooth pieces of the boundary $\partial Q$ meet at $q$, and is {\it
tangential} if the velocity $v$ is tangential to $\partial Q$ at $q$. We shall
denote the set of all such singular reflection points (belonging to any of the
above two types, in case of multiple collision supplied with outgoing velocity
$v^+$) by ${\cal SR}^+$. We introduce some further notation related
to the singularities. $M^*\subset M$ is the set of phase
points whose trajectories contain infinitely many collisions such that at most one is singular among them. $M^0\subset M^*$ is the set of
regular phase points --- i.e. whose entire trajectory (with infinitely many
collisions on it) avoids ${\cal SR}^+$; for the points in $M^1=M^*\setminus M^0$ there is exactly one singular
reflection. To measure the size of these sets we use the small inductive topological dimension (for details see \cite{gomb}). It is not hard to see that ${\cal SR}^+$ is a codimension 1 subset
of $\partial M$, and, as a consequence, $M^0$ is of full measure in $M$, while
$M\setminus M^*$ is of codimension 2 (cf. \cite{alapt}). More precisely, it
may happen that some trivial 1-codimensional subsets are present for the
points of which the (semi-)trajectories do not collide at all (see
subsection~\ref{secEdyn} in this paper or \cite{ort}).
The main philosophy of the study of semi-dispersive billiards has always been
the principle that in some sense {\it hyperbolic behaviour is stronger than
the effect of singularities}. According to this philosophy we expect that
dynamics in a cylindric billiard is chaotic and ergodic unless some geometric
degeneracy of the cylinders is present. This expectation is formulated
precisely in the conjecture below, for which we need some more preparation.
For any cylinder $C_i$ let ${\cal G}_i$ be the group of all orientation preserving orthogonal
transformations in ${\bf R^d}$ that leave the points of the
generator subspace $A_i$ fixed. Denote furthermore by ${\cal G}$ the group of
transformations algebraically generated by all such groups ${\cal G}_i: i=1,...,k$. Observe that the group ${\cal G}$, being a subgroup of the special orthogonal
group $SO(d)$, has a natural action on ${\bf S^{d-1}}$ (and thus on ${\bf R^d}$
as well).
\begin{conjecture}
The cylindric billiard is completely hyperbolic and moreover ergodic if and only if the
group action of ${\cal G}$ is transitive on the unit velocity sphere ${\bf
S^{d-1}}$. From here on we refer to the transitivity of the action of ${\cal
G}$ as to the transitivity condition on our cylinders.
\label{Ctr}
\end{conjecture}
The remarks below are discussed in detail in \cite{trans}.
\begin{remark}
For the case of restricted hard ball interactions (see the example above) the transitivity
condition for the isomorphic cylindric billiard is satisfied if and only if the collision
graph is connected. Thus if one were able to prove the above conjecture one
would immediately get ergodicity and hyperbolicity for a large class of
particle systems with hard ball interactions, including the so much discussed Hard
Ball System case.
\label{Rtrg}
\end{remark}
\begin{remark}
The equivalence of the following three conditions on the cylindric billiard is
demonstrated in \cite{trans}: \\
(i) The action of ${\cal G}$ is transitive on ${\bf S^{d-1}}$ \\
(ii) The action of ${\cal G}$ is irreducible on ${\bf R^d}$ \\
(iii) The system of base subspaces $L_1, \cdots ,L_k$ has the Orthogonal
Non-Splitting Property, i.e. there is no orthogonal splitting ${\bf R^d}=K_1
\oplus K_2$ for which $dim(K_j)>0$ and which has the property that for any
$i=1, \cdots ,k$ either $L_i \subset K_1$ or $L_i \subset K_2$.
\label{Rtrn}
\end{remark}
As a consequence of remark~\ref{Rtrn} we immediately see that {\it the
transitivity condition is necessary} for both ergodicity and hyperbolicity. Indeed, let us suppose that the transitivity condition does
not hold. This in turn implies by virtue of (iii) above that there exists a
nontrivial orthogonal splitting ${\bf R^d}=K_1 \oplus K_2$. Then it is easy to
see that for any phase point $(q,v) \in M$ the two quantities
$\| P_{K_j}(v)\|$ remain constant under the time evolution, thus nontrivial integrals
of motion are present (here and from here on $P_K(v)$ stands for the
orthogonal projection of the vector $v$ onto the subspace $K$, while $\| z\|$
is the absolute value of the vector $z$).
In contrast to the necessity of the condition, proving that the transitivity
of the action is indeed sufficient for both ergodicity and hyperbolicity is an
extremely nontrivial task. (By remark~\ref{Rtrg}, proving conjecture~\ref{Ctr}
in its full generality would imply the ergodicity and the hyperbolicity of
Hard Ball Systems, a problem that has been a subject of active research for
the last couple of decades.) The results in this paper are proofs of the
conjecture for some subclasses of cylindric billiards. In the rest of this
subsection we give a summary of results.
The simplest possible class of cylindric billiards one could imagine would be
the case $A_i=\{ 0\}$ for all the cylinders $C_i$. For such systems the
scatterers are strictly convex, the billiard is dispersive, thus ergodicity
and hyperbolicity follow as consequences of the local hyperbolic and ergodic
theorems (see \cite{alapt} or \cite{chs}). On the other hand the transitivity
condition is naturally satisfied. \\ To get more complicated cylindric
billiards we must allow for `thicker' generator subspaces. The more cylinders
may have generator subspaces with nontrivial intersection, the more difficult
it is to handle the system. Theorem~\ref{Te} below proves the desired properties for the simplest
possible nontrivial class, while Theorem~\ref{Th} gives the hyperbolicity of
some more 'complicated' systems. In a sense the complexity of the increasing
number of generator subspaces with nontrivial intersection corresponds to the
increasing number of particles in Hard Ball Systems. Indeed, the system of two balls
on the torus is equivalent to a dispersive billiard (cf. \cite{alapt});
Theorem~\ref{Te} is valid for the system of three balls; while adaptation of
the proof for Theorem~\ref{Th} gives hyperbolicity for four balls (see the
Appendix).
\begin{theorem}
Let us consider a cylindric billiard with an arbitrary number of scatterers
$C_1, \cdots ,C_k$ \\ (i) which satisfy the transitivity condition;\\ (ii)
for which it is true that for any two cylinders $C_i, C_j \; (i\ne j)$ the corresponding generator subspaces are transversal:
$A_i\cap A_j =\{ 0 \}$. \\ The dynamical system is ergodic and
hyperbolic. (Moreover, by \cite{ks} and \cite{hch}, the dynamics is K-mixing and
possesses the Bernoulli property).
\label{Te}
\end{theorem}
{\bf Remarks.} This result is a natural generalization of the one discussed in
\cite{2heng}: the ergodicity and hyperbolicity of a billiard with two
cylinders was demonstrated there, with one additional assumption on the
scatterers besides our (i)-(ii). \\ The result of Theorem~\ref{Te} implies the
ergodicity of three particles, both for the classical case of
three hard balls with positive radii and for the case when two of the three
particles interact only with the third particle and not with each other as
they have zero radius.
\begin{theorem}
Let us consider a billiard with three cylinders which satisfy the
transitivity condition. \\ The dynamical system is
hyperbolic.
\label{Th}
\end{theorem}
{\bf Remarks.} As for a particle system that belongs to this
class one may consider four particles only one of which has nonzero radius (in
this model the particles with zero radius do not interact; we have three pair
interactions, i.e. three cylinders).\\ It is a natural question what one can
do if all the four particles have nonzero radius. A slight adaptation of our
methods gives hyperbolicity for this case as well (as it is demonstrated in
the Appendix). Although for four hard balls even ergodicity has been proven
(\cite{4golyo}), the new results are interesting as they are valid without
dimensional restriction, i.e. for two dimensional disks as well.
One more remark is in order. It may happen that two cylinders are parallel, i.e. for a pair $i\ne j$ $A_i=A_j$. In such a case collisions with $C_i$ and $C_j$ have exactly the same effect on the dynamics, thus the two cylinders can be considered as identi
cal. We have formulated the theorems above and will go on with the proofs below by assuming that such a parallelity does not occur. Nevertheless the results remain trivially true if we state the conditions on such identified classes of parallel cylinders
rather the on the cylinders themselves.
\subsection{Basic strategy of the proofs}
\label{secAltpr}
In this subsection some 'traditional' concepts are summarized that play an
essential role in the theory of semi-dispersive billiards. Our discussion is
very brief, for more details see the literature, especially
\cite{trans,alapt,4golyo}. Let us consider a nonsingular finite trajectory segment
$S^{[a,b]}x$, where $a<0**0) s.t. \forall
\alpha \in (-\delta,\delta)\\
& & p(S^a(q(x)+\alpha w, v(x)))=p(S^ax) \& \\
& & p(S^b(q(x)+\alpha w, v(x)))=p(S^bx) \}.
\end{eqnarray*}
Observe that $v(x) \in {\cal N}_0 (S^{[a,b]}x)$ is always true, the
neutral subspace is at least $1$ dimensional. Neutral subspaces at time
moments different from $0$ are defined by ${\cal N}_t (S^{[a,b]}x):={\cal N}_0 (S^{[a-t,b-t]}(S^tx))$, thus they are naturally isomorphic to the one at $0$.
The following notion is one of the most important concepts in the theory of
semi-dispersive billiards. The non-singular trajectory segment $S^{[a,b]}x$ is {\bf
sufficient} if for some (and in that case for any) $t \in [a,b]: \quad
dim({\cal N}_t (S^{[a,b]}x))=1$. A point $x \in M^0$ is said to be sufficient
if its entire trajectory $S^{(-\infty,\infty)}x$ contains a finite sufficient
segment. Singular points are treated by the help of trajectory branches (see \cite{alapt}): a point $x\in M^1$ (this precisely means that the entire trajectory
contains one singular reflection) is sufficient if both of its trajectory
branches are sufficient.
Sufficiency has a picturesque meaning; roughly speaking a trajectory segment
is sufficient if it has encountered all degrees of freedom. Nevertheless the
concept is important as very strong theorems hold in open neighborhoods of sufficient points (on
more general formulations of sufficiency and on the local ergodicity theorem
see \cite{lw,chs,alapt} ).\\ {\bf Local Hyperbolicity Theorem.} Every
sufficient phase point $x \in M$ has an open neighborhood $x\in U\subset M$, such
that for $\mu$ a.e. $y\in U$ the relevant Lyapunov exponents of the flow are
nonzero. \\ {\bf Local Ergodicity Theorem or Fundamental Theorem of
Semi-Dispersive Billiards.} Assume that some geometric conditions are true for
the singularities of our semi-dispersive billiard (most importantly the {\it
Chernov-Sinai Ansatz} holds, see e.g. \cite{alapt}). Then every sufficient
phase point $x \in M$ has an open neighborhood $x\in U\subset M$ which belongs to
one ergodic component.
Now let us examine the trajectory segment $S^{[a,b]}x$ from another point of
view. We denote with $\tau_j$; $a<\tau_1< \cdots <\tau_n ****0$ such that the collision sequence of $S^{[t,\infty)}$ contains only one archipelago.
\begin{proposition}
For every cell $R$ of maximal dimension $2d-3$ in ${\cal SR}^+$, the set $R_{es}\subset R$ of eventually simple points can be covered by countably many closed zero-subsets (with respect to the induced measure $\mu_C$) of $C$.
\label{Psing}
\end{proposition}
{\bf Proof} of this Proposition, just like as it was with the analogous
statements in \cite{2heng,ort}, is an adaptation of the proof of
Main Theorem 6.1 in \cite{4golyo}. Let us denote one particular eventually simple
collision sequence (containing only one archipelago) by $\Sigma$. There are two possible cases, either $\Sigma\sim (a)$ or $\Sigma\sim (a,b)$ with $L_a\perp L_b$. For both of them there is definitely at least one
cylinder $C_c$ avoided by the semi-trajectory $S^{[t,\infty)}$. In other words our
eventually simple phase point belongs to a ball avoiding set $F_+$ (the
definitions of $y_0,\; U(y_0),\; F_+,\; F_+'$ are taken from the above
references). We also define the pseudo-stable invariant manifolds $\gamma^{sp}_0(y)$
and $\gamma^s_e(y)$ according to the literature: if $\Sigma \sim (a)$, we have
$dim(\gamma^{sp}_0(y))=dim(A_a)$ and $dim(\gamma^s_e(y))=dim(L_a)-1$; while
for $\Sigma \sim (a,b)$ we get $dim(\gamma^{sp}_0(y))=1$ and
$dim(\gamma^s_e(y))=dim(L_a)+dim(L_b)-2$. However in both cases we arrive,
just like in the references, at the $d-1$-dimensional $\gamma^s_g(y)$,
which is a concave orthogonal manifold. The key point is again that backward
images of concave orthogonal manifolds are always transversal to the set of
singular reflections (as it is true by sublemma 4.2 in \cite{alapt}). This
together with weak ball avoiding gives an indirect proof of the Proposition
(similarly to the arguments in Lemma~\ref{LEdyn}). $\Box$
Propositions~\ref{P2kd}, ~\ref{PEdyn} and ~\ref{Psing} altogether imply that any connected component of the set $M^\#$ belongs to one ergodic component. As already mentioned at the first paragraph of the subsection, a reference to Lemma A.2.3 from \cite{ort} finishes the proof of the required global ergodicity.
\section{Theorem~\ref{Th} -- three cylinders}
\label{secHyp}
\setcounter{equation}{0}
In this section our aim is to prove, following the steps (H1)-(H2) mentioned in subsection~\ref{secAltpr}, the hyperbolicity of a billiard with three cylinders which satisfy the transitivity condition of conjecture~\ref{Ctr}. In symbolic collision sequenc
es we refer to the three cylinders $C_1, C_2, C_3$ simply by the numbers 1, 2 and 3. In all our notation we follow section~\ref{secErg}, there is only one difference: to simplify the calculation of neutral subspaces in subsection~\ref{secHalg}, we adopt a
convention from \cite{4golyo}; we fix the advance of one particular (central) collision zero. That way sufficiency of the segment is equivalent to the triviality of the neutral subspace, ${\cal N}=\{ 0\}$.
The specific {\bf notion of richness} we use for our model is exactly the
double connectedness already mentioned in subsection~\ref{secAltalg}: a phase
point $x$ is rich if there exists a cutting time moment $t$ on its
entire trajectory, such that the symbolic collision sequences (in their
shortest possible equivalent form) for both segments $S^{(-\infty ,t]}x$ and
$S^{[t,\infty )}x$ are connected (i.e. they contain enough cylinders to
generate a transitive action). \\
The essence of our geometric-algebraic considerations
(subsection~\ref{secHalg}) is to prove
\begin{proposition}
Assume $x\in M^0$ is an arbitrary rich and regular phase point. There exist an open neighborhood $x\in U_0$ and a submanifold $N$ of $M$ such that (i) $codim(N)\ge 1$ and (ii) $\forall y \in U_0 \setminus N$ is sufficient.
\label{PHalg}
\end{proposition}
As for the dynamical-topological part our key statement is
\begin{proposition}
Let us denote by $M_p^0$ the set of regular and poor phase points. The set $M_p^0$ is of zero $\mu$-measure.
\label{PHdyn}
\end{proposition}
Actually in subsection~\ref{secHdyn} we prove a stronger statement (Lemma~\ref{LHdyn}) which implies the Proposition above. We finish the discussion of Theorem~\ref{Th} in the same subsection.
\subsection{Geometric-Algebraic Considerations}
\label{secHalg}
In the case of three cylinders one easily gets a classification of all
possible rich collision sequences. Indeed, a rich collision sequence
necessarily contains a subsequence $\Sigma$ which has (up to equivalence) one
of the forms below:\\
(a) $\Sigma \sim (1,(2-3)_{\ge 3},1)$ (here $(2-3)_{\ge 3}$ stands for a sequence of the cylinders $C_2$ and $C_3$ which contains at least three consecutive collisions in the shortest equivalent form); \\
(b) $\Sigma\sim ((2-3)_{\ge 3},1,3)$; \\
(c) $\Sigma\sim (a,b,c)$ with $A_a\cap A_b=A_a\cap A_c=A_b\cap A_c=\{ 0\}$ (the possibility $a=c$ is not excluded); \\
(d) $\Sigma\sim (1,2,3,1)$ with $A_1\cap A_2=A_1\cap A_3 =\{ 0\}$ but $A_2\cap A_3\ne\{ 0\}$; \\
(e) $\Sigma\sim (1,2,3,1,2)$ with $A_1\cap A_3\ne\{ 0\}$\\
This classification can be obtained if we distinguish between two
possibilities; if in the first connected part of the sequence all the three
cylinders appear, we are led to one of the cases (a), (b) or (e); if on the
contrary, only two cylinders are present in the first connected part, cases
(c) and (d) may occur as well.
Now our task is to describe the one-codimensional submanifolds $N$ for all the
above five cases.
{\bf Case (a).} Non-collision time moments directly after the first and before
the second collision with $C_1$ are denoted by $t_-$ and $t_+$, while the
advances of these two collisions are $\alpha$ and $\alpha'$, respectively. We
shall get $N=N^{(1)}\cup N^{(2)}$ with both $N^{(1)}$ and $N^{(2)}$ one-codimensional. Observe
that we may exclude the possibility $L_2\perp L_3$, otherwise we would have a
shorter equivalent form.
To get $N^{(1)}$ consider the trajectory segment $S^{[t_-,t_+]}x$. Time
evolution in this shorter time interval is determined by the sub-billiard
dynamics defined by cutting out only two cylinders $C_2$ and $C_3$ (this can
be understood on the torus ${\bf T^{d'}}$ where $d'=dim(L_2+L_3)$, together
with almost periodic motion in the directions of $A_2\cap A_3$). From
Proposition~\ref{P1kd} we however know that, in a small open neighborhood of $x$
phase points non-sufficient with respect to this sub-dynamics in the time
interval $[t_-,t_+]$ lie in a one-codimensional submanifold $N^{(1)}$ (remember
$L_2\not\perp L_3$ !).
Consider now on the contrary phase points which do not lie on $N^{(1)}$,
i.e. which are sufficient with respect to the above mentioned
subdynamics. Among them we would like to characterize those non-sufficient in
the whole time interval $[a,b]$ for the full billiard dynamics. We shall show
below that such points form a one-codimensional submanifold $N^{(2)}$. The
advances of all the collisions with $C_2$ and $C_3$ are equal, so we may use
the convention mentioned in the beginning of section~\ref{secHyp} and fix the
advances for all these central collisions to be zero. Observe that for any
neutral vector calculated at time moments $t_-$ and $t_+$:
$w^-=w^+=w \in A_2\cap A_3$. Moreover, neutrality with respect
to the first and the second collisions with $C_1$ implies (by (\ref{1heng}))
$$
\alpha v^-_1=w^-_1=w^+_1=\alpha' v^+_1
$$
As $\alpha=0$ or $\alpha'=0$ would mean sufficiency we may conclude that
\be
v^-_1 \| v^+_1
\label{Hca}
\ee
Apply now all possible purely configurational translations $\delta q$ at time
moment $t_-$; $v^-$ does not change while the velocity difference $v^--v^+$
moves on a surface of a sphere of highest possible dimension in
$L_2+L_3$. Following the argument at the end of case 1 from
Proposition~\ref{P1kd}, we see that the perturbations $\delta q$ definitely
give a direction transversal to the submanifold $N^{(2)}$ of non-sufficient points.
($L_1\perp (L_2+L_3)$ is not possible, in that case the transitivity condition
would not hold for our three cylinders). Thus $codim (N^{(2)}) \ge 1$.
The discussion of {\bf Case (b)} is analogous to Case (a) and we leave it to the reader (see also cases 9 and 11 in section 4 of \cite{4golyo}). In {\bf Case (c)} we may directly refer to Proposition~\ref{P1kd} to see that the submanifold $N$ is indeed (a
t least) one-codimensional.
As to {\bf Case (d)} the key is again reference to Proposition~\ref{P1kd} with
some adaptation. Notation for the advances is the following: $\alpha$ for the
first collision with $C_1$, $\beta$ for the collision with $C_2$ and $\alpha'$
for the second collision with $C_1$. Our central collision the advance of
which we fix as zero is the one with $C_3$. Non-collision time moments $t_-$,
$t_0$ and $t_+$ are fixed directly after collisions with $C_1$ (for the first
time), $C_2$ and $C_3$, respectively.
We may assume that any two of the three base subspaces are
transversal. Indeed, otherwise we would get (see case 2 of
Proposition~\ref{P1kd}) that, apart from a 1-codimensional submanifold $N_1$,
a subsegment $(i,j)$ of two consecutive collisions is sufficient with respect
to the sub-billiard defined by the two cylinders. Thus we would be in one of
the positions of case (a) or (b). By the same argument we may exclude $\beta=0$
and $\alpha'=0$. The assumption that both $L_1\not\perp L_2$ and $L_1\not\perp
L_3$ is possible. Indeed, in the presence of such an orthogonality our
sequence would not be rich (and, continuing it in any way to
a rich sequence would lead to one of the cases (a) or (b) again).
If $L_2\perp L_3$, the argument from case 3 in Proposition~\ref{P1kd} goes
through with standard modifications (the argument should be 'projected down
orthogonally' onto $L_2+L_3$). The possibility $L_2\not\perp L_3$, which we
discuss in a bit more detail, is similar to case 1 in
Proposition~\ref{P1kd}. By the zero advance of the central collision for any
neutral vector at time moments $t_0$ and $t_+$: $w^0=w^+=w \in A_3$. On the
other hand by (\ref{1heng}):
\be
w_1=\alpha' v^+_1, \quad w_2=\beta v^0_2
\label{Hcd}
\ee
Let us denote by $J$ the linear mapping of ${\bf R^d}$ onto itself which sends any vector into the sum of its orthogonal projections onto $L_1$ and $L_2$, $Jz=z_1+z_2$ (this is a linear bijection by $A_1\cap A_2=\{ 0\}$). Summing the two equations in (\ref{Hcd}) and applying $J^{-1}$ we get for non-sufficient points:
\be
\exists\lambda\in {\bf R}; \lambda\ne 0: \quad J^{-1}(\lambda v_2^0 +v_1^+) \in A_3
\label{Hcd1}
\ee
Now apply all possible purely configurational translations $\delta q$ at $t_+$, $v^+$ remains the same while $v^0$ changes into $\bar{v}^0$. Assume (\ref{Hcd1}) is true for $\bar{v}^0$ with some $\bar{\lambda}\in {\bf R}$, then:
$$
J^{-1}(\lambda v_2^0- \bar{\lambda}\bar{v}^0_2) \in A_3
$$
However it is only true for vectors inside $A_1$ that their images under $J$ lie inside $L_2$. This, together with the bijectivity of $J$ and the assumed $A_1\cap A_3=\{ 0\}$, gives:
$$
\lambda v_2^0= \bar{\lambda}\bar{v}^0_2 \; \Leftrightarrow \; v_2^0 \| \bar{v}^0_2
$$
Just like in case 1 from Proposition~\ref{P1kd}, what really matters is that orthogonality is not allowed, $L_3\not\perp L_2$. We may find, following exactly the argumentation there, a direction transversal to $N$, and thus conclude $codim(N)\ge 1$.
In all the above cases we could describe the one-codimensional manifold $N$ with the help of former methods (in cases (c) and (d) by techniques from subsection~\ref{secEalg}, in cases (a) and (b) by adaptation of the proofs from \cite{4golyo} to the cylin
dric billiard setting). The situation is however quite different with {\bf Case (e)}.
\begin{lemma}
Consider a phase point $x$ with collision sequence $\Sigma \sim (1,2,3,1,2)$,
where $A_1\cap A_3\ne\{ 0\}$ (i.e. case (e)). The
statement of Proposition~\ref{PHalg} is true for $x$.
\label{LHalg}
\end{lemma}
{\bf Proof.} We may assume that neither orthogonality, nor intersection is
present for any pair of base subspaces. Indeed, in case $L_i\perp L_j \;
(i\ne j)$ our collision sequence would be equivalent to any of the above cases
(a) or (b). On the other hand if $L_i\cap L_j\ne\{ 0\}$ for some $i\ne j$,
just like as it was discussed at the beginning of case (d) above, we would
have sufficiency in a sub-billiard for the subsegment $(i,j)$ outside a
1-codimensional manifold $N^{(1)}$, and a reference to the arguments in cases (a)
or (b) would give $N=N^{(1)} \cup N^{(2)}$.
We choose 3 as our central collision and fix the advance as zero. Notation for the other advances:\\
$\alpha'$ for the first collision with $C_1$ and $\alpha$ for the second;\\
$\beta$ for the first collision with $C_2$ and $\beta'$ for the second.\\
We exclude the possibilities $\alpha=0$ and $\beta=0$, in these cases we would
have sufficiency in a sub-billiard for one of the sequences $(3,1)$ or $(2,3)$
and could repeat the argument above.
The non-collision time moments we fix are:\\
$t_*$ after the first 1 and before the first 2;\\
$t_-$ after the first 2 and before 3;\\
$t_+$ after 3 and before the second 1;\\
$t_{\#}$ after the second 1 and before the second 2.\\
Depending on the geometrical position of the cylinders we distinguish two further subcases.
{\bf Subcase (e1).} $dim(L_1\cap A_2)\le 1$, thus the two subspaces are either transversal or the intersection $L_1\cap A_2$ is a line. By the convention of zero advance at the central collision we know that for any neutral vector at time moments $t_-$ an
d $t_+$, $w^+=w^-=w\in A_3$. Moreover, by applying (\ref{cpf}) to our collision sequence we get the set of equations:
\begin{eqnarray}
\nonumber
\alpha v^+_1 &=& w_1\; =\; (\alpha'-\beta)v^*_1 + \beta v^-_1\\
\beta v^-_2 &=& w_2\; =\; (\beta'-\alpha)v^{\#}_2 + \alpha v^+_2
\label{Hcecpf}
\end{eqnarray}
The first equation implies, that for non-sufficient points:
\be
\exists\lambda\in{\bf R}; \lambda\ne 0:\quad (v^+_1-\lambda v^-_1)\, \| v^*_1
\label{Hcemozg}
\ee
while from the second equation in (\ref{Hcecpf}) we get that {\it with the same} $\lambda$:
\be
(\lambda v^-_2-v^+_2)\, \| v^{\#}_2
\label{Hcefix}
\ee
Now assume for a while that neither $v^-_2 \| v^+_2$ nor $v^-_1 \|
v^+_1$ and apply a purely configurational translation at time moment $t_-$ for
which $\delta q \in A_1\cap A_3$. Under the effect of such a perturbation:\\
(i) {\it none of the velocity vectors $v^-$, $v^+$ and $v^{\#}$
changes}. Thus if the perturbed phase points remain non-sufficient, as the
vector quantities in (\ref{Hcefix}) do not change, equation (\ref{Hcemozg})
holds throughout the perturbation with the original constant $\lambda$
(remember that parallelity is excluded).\\
(ii) On the other hand as $A_1\cap A_2\cap A_3=\{ 0\}$ (otherwise our
cylinders could not satisfy the transitivity condition), under the effect of
the above perturbation {\it the velocity vector at time moment} $t_*$ {\it
changes},
\be
v^* \; \rightarrow \; \bar{v}^*
\label{Hcepert}
\ee
where the velocity difference $v^* -\bar{v}^*$ moves on an arc of a circle
that goes through the origin in $L_2$. (More precisely, if $dim(A_1\cap
A_3)=1$, it might be possible that for such a perturbation $\delta q \in
A_1\cap A_3$ the velocity $v^*$ does not change. However in that case $v_2^-
\in ((A_1\cap A_3)+A_2)$. Thus the (at least) two dimensional velocity
component $v^-_2$ is restricted to a line, which means a one-codimensional
restriction for our phase point.)
Now we finish our argument similarly to case 1 from
Proposition~\ref{P1kd}. Observe that by (i), as the left hand side of
(\ref{Hcemozg}) is the same for $v^*$ and $\bar{v}^*$:
\be
v^*_1\| \bar{v}^*_1
\label{Hcekd}
\ee
If $L_1\cap A_2=\{0 \}$, then if we project orthogonally the arc of the
perturbation
(\ref{Hcepert}) onto $L_1$ it remains an ellipse, thus (\ref{Hcekd}) cannot
hold. If $L_1\cap A_2\ne \{0 \}$, then by the nature of dynamics
$$
P_{L_1\cap A_2}(v^*)=P_{L_1\cap A_2}(v^-)=P_{L_1\cap A_2}(\bar{v}^*)
$$
If $P_{L_1\cap A_2}(v^-)\ne 0$ (otherwise we have velocity restriction giving $codim(N)\ge 1$ itself), we know that (\ref{Hcekd}) can only hold if $v^*_1=\bar{v}^*_1$, which is impossible
as in subcase (e1) we assumed $dim(L_1\cap A_2)\le 1$.
Finally let us say a few words about the points for which e.g. $v^-_1 \|
v^+_1$. Apply all possible purely configurational translations at $t_-$; $v^-$
does not change so for the perturbed velocity at time $t_+$ we expect
$\bar{v}^+_1 \| v^+_1$. However $\bar{v}^+-v^+$ moves on a surface of a sphere of
full dimension inside $L_3$, and, after projecting orthogonally onto $L_1$, we
get (by $L_1\not\perp L_3$): $codim{N}\ge 1$ exactly the same way as at the end
of case 1 from Proposition~\ref{P1kd}. (Observe that the condition
$dim(L_1\cap A_2)\le 1$ of subcase (e1) was not used in the analysis of this
parallelity.)
{\bf Subcase (e2).} $dim(L_1\cap A_2) \ge 2$. At first we discuss what we can do
if $(L_1\cap A_2)\not\perp L_3$ (in other words if $L_1\cap A_2\not\subset
A_3$). By the nature of dynamics for any neutral vector $w$ we get the following series of equations:
$$
P_{L_1\cap A_2}(w)=P_{L_1\cap A_2}(w^*)=\alpha' P_{L_1\cap A_2}(v^*)=\alpha'
P_{L_1\cap A_2}(v^-)
$$
Here $w^*$ is the value of the neutral vector $w$ at time moment $t_*$. On the
other hand trivially:
$$
P_{L_1\cap A_2}(w)=\alpha P_{L_1\cap A_2}(v^+)
$$
Thus if the point is not sufficient:
$$
P_{L_1\cap A_2}(v^-) \| P_{L_1\cap A_2}(v^+)
$$
Apply all possible configurational translations at $t_-$ to conclude, exactly
the same way as with the parallelity discussed at the end of subcase (e1), that
$codim(N) \ge 1$.
Problem only arises with the above argument if $L_1\cap A_2 \subset A_3$, a
possibility we discuss in an indirect way. First of all observe that in such a
case there are a couple of generalizations for the main argument of subcase
(e1). By $L_1\cap A_2 \subset A_3$ trivially $A_2\cap A_3 \ne \{ 0\}$. Thus we
may apply the whole discussion word by word for the reverse directed sequence
with $1\leftrightarrow 2$, (in that case we would e.g. apply $\delta q\in
A_2\cap A_3$ at time $t_+$). It is also
true that the scheme works for higher dimensional $L_1\cap A_2$ as well
whenever
\be
dim(L_1\cap A_2) \le dim(A_1\cap A_3)
\label{Hceng}
\ee
Indeed, following the argument of
subcase (e1), after application of $\delta q \in A_1\cap A_3$ at time moment
$t_-$, the dimension of the sphere on the surface of which the velocity
difference $v^*-\bar{v}^*$ moves inside $L_2$ is
$min(dim(A_1\cap A_3), dim(L_2)-1)$ (otherwise we would have a
restriction on the velocity $v^-$, which would mean $codim(N)\ge 1$
itself). Problem only arises if there are too many orthogonal directions
between $L_1$ and $L_2$, in other words if (\ref{Hceng}) does not hold
(otherwise the effect of our perturbation is 'visible' at the orthogonal
projection of our velocity difference onto $L_1$.)
Assume now that $dim(L_1\cap A_2)=k\ge 1$. However, from $L_1\cap A_2 \subset
A_3$ trivially follows that $dim(A_2\cap A_3)\ge k$, thus by the above
considerations (now applied with $1\leftrightarrow 2$) there is no problem if
\be
dim(L_1\cap A_2) \ge dim(L_2\cap A_1)
\label{Hceng1}
\ee
Now we are ready with our indirect proof with one more reference to the $1
\leftrightarrow 2$ symmetry as either (\ref{Hceng1}) or the opposite
inequality surely holds.
The demonstration of the Lemma is ready. $\Box$. Hence the
Proposition. $\Box$.
We close this subsection with an {\bf example}. The strange geometric position
of subspaces discussed in subcase (e2) above might seem quite unnatural (such a
thing does not happen in hard ball systems, see also the appendix). One can
however easily find an example from a well studied category, namely an
``orthogonal cylindric billiard'' (see \cite{ort} as basic reference for this
example). In orthogonal cylindric billiards the base subspaces for the
cylinders are given by subsets of a fixed orthonormal basis in ${\bf
R^d}$. We define our billiard on ${\bf T^8}$, the base subspaces are
$$
K_1=\{ 1,2,3,4\}; \; K_2=\{ 3,4,5,6\}; \; K_3=\{ 6,7,8\}.
$$
For this model $dim(L_1\cap A_2)=2$ and $L_1\cap A_2 \subset A_3$.
\subsection{Dynamical-Topological Considerations and finishing the proof}
\label{secHdyn}
Proposition~\ref{PHdyn} is a direct consequence of the following Lemma:
\begin{lemma}
Denote by $M^0_e$ the set of those regular phase points $x$ for which there
exists a time moment $t_0$ such that the segment $S^{[t_0,\infty)}x$ is not
connected. $\mu(M^0_e)=0$.
\label{LHdyn}
\end{lemma}
{\bf Proof} of this Lemma, as it might not be surprising for the reader, is
again weak ball-avoiding (\cite{wart,gomb}). Without loss of generality we may
restrict our attention to points $x\in M^0_e$ for which $S^{[0,\infty)}x$ is
not connected. Now by the orthogonal splitting (see remark~\ref{Rtrn})
dynamics for this semi-trajectory is determined by a product dynamics of
K-system(s) and almost periodic motion (the
'K-factors' in the product dynamics are either dispersive billiards or K
according to section~\ref{secErg}). On the other hand the positive
semi-trajectory avoids one of the cylinders, e.g. $C_3$. However, for any of
the product dynamics above weak ball-avoiding theorems can be applied, thus
indeed $\mu(M^0_e)=0$. $\Box$
Propositions~\ref{PHalg} and ~\ref{PHdyn} together with the local
hyperbolicity theorem (see subsection~\ref{secAltpr}) give the proof of
Theorem~\ref{Th}. Full hyperbolicity (the fact that the Lyapunov exponents are
nonzero almost everywhere) implies, on the basis of Katok-Strelcyn theory
(\cite{ks}), that the system has at most a countable number of ergodic
components, each of positive measure. Moreover, on any of these ergodic
components the dynamics is K-mixing (see e.g. \cite{nandi}) and possesses the
Bernoulli-property (see e.g. \cite{hch}).
\bigskip \bigskip
{\bf Concluding Remarks.} Of course the natural question that arises how we
could generalize the results of Theorems~\ref{Te} and ~\ref{Th} further. If we
increase the number of cylinders, we encounter more and more difficulty
in the {\it geometric-algebraic considerations} as the number of collision
sequences to be studied (with, moreover, different possible geometric
positions of the cylinders) is much higher. This makes things especially
complicated if we want to prove ergodicity, thus we should 'gain' two
codimensions from the equations on the neutral subspaces. In a general
setting (just like in our Proposition~\ref{P2kd}) one may hope to get
$codim(N)\ge 2$ by $N=N_1 \cap N_2$ with $N_1$ and $N_2$ transversal and both
one-codimensional. However, as the number of possible collision sequences
increases, there are more and more possibilities for the manifolds $N_1$ and
$N_2$ above, in many cases obtained in quite an implicit way (see
Lemma~\ref{LHalg}), thus proving transversality for all cases seems very
difficult, if not impossible (even in the setting of Theorem~\ref{Te} one has
to consider several possibilities, see Proposition~\ref{P2kd}). I think that
proving the ergodicity part of conjecture ~\ref{Ctr} is at the present level
much far away; we might not even guess what the suitable notion of richness
for the proof of ergodicity could be. However for {\it proving hyperbolicity}
even in a general setting {\it the notion of richness (double connectedness)
we use in section}~\ref{secHyp} {\it seems to be enough}. Nevertheless this
task is highly nontrivial at the geometric-algebraic part.
As to the {\it dynamical-topological considerations}, although results from
the literature were quite easily adapted to our setting, if we increase the
number of cylinders this does not remain true. The main problem is that ball
avoiding theorems, in their classical form, are {\it of inductive nature},
i.e. they rely on the K-property of smaller subsystems. For Hard Ball Systems a new type
of ball avoiding theorem is used in \cite{ngolyo}, which is not inductive in
its nature. This method, however, uses special symmetries of hard balls, thus
its adaptation to the general cylindric billiard setting seems impossible.
Altogether we can say that if we want to discuss ergodic and hyperbolic
properties for all cylindric billiards, i.e. to prove conjecture~\ref{Ctr} in
its full generality, a kind of breakthrough would be needed. Nevertheless the
simple Lemma~\ref{Lcpf} could be a good starting point (for the
geometric-algebraic considerations) even in such a general setting. We could
say that {\it this Lemma 'saves' as much of the connected path formula} -- which
uses, e.g., the conservation of momenta in hard ball systems -- {\it for the
cylindric billiard setup as possible} (see also the Appendix). Probably the
most remarkable consequence of our results is that they prove chaotic/ ergodic
properties for full subclasses of cylindric billiards. Thus {\it we may get more convinced about the
validity of conjecture}~\ref{Ctr}. In other words there is
increasing evidence that the Boltzmann-Sinai Ergodic Hypothesis is true
because Hard Ball Systems belong to the class of cylindric billiards that satisfy the
transitivity condition.
\section*{Appendix: Some remarks on Hard Ball Systems}
\label{secA}
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The most studied and physically most interesting cylindric billiards are
particles with some hard ball pair-interactions, especially the system of Hard
Balls. Though these systems possess some remarkable symmetries, their study
gets more and more complicated as we increase the number of interacting
balls. Ergodicity of two balls in any dimensions follows directly from the
fundamental theorem as this system is a dispersive billiard
(\cite{chs,alapt}). Three balls are only semi-dispersive, thus their
ergodicity is a much more difficult task (\cite{3golyo}). Generalization to
the case of four balls has been achieved only with the dimensional restriction
$\nu \ge 3$ on our balls (\cite{4golyo}). Later on even the ergodicity of an
arbitrary number of hard balls has been proven (\cite{nandi}), however the
dimensional restriction $\nu\ge N$ is needed.
As for the hyperbolicity of Hard Ball Systems, applying highly nontrivial
techniques in the geometric-algebraic part Sim\'anyi and Sz\'asz managed to
discuss all possibilities for $N$ and $\nu$. Nevertheless the result in
\cite{ngolyo} only holds apart from a countable union of analytic submanifolds
for the outer geometric parameters (i.e. for the masses and radii of the
particles).
Ergodicity and/or hyperbolicity {\it for some classes of hard ball systems}
follows from the results of this paper. Any system of three interacting
particles (with a connected collision graph) is a special case of the billiard
discussed in section~\ref{secErg} (see also subsection~\ref{secAltre}). Thus
Theorem~\ref{Te} can be viewed in a way as a generalization of \cite{3golyo}
to a
cylindric billiard setting. A nice example of a Hamiltonian system with
restricted hard ball interactions
that belongs to the class of cylindric billiards discussed in
section~\ref{secHyp} is the one already mentioned in
subsection~\ref{secAltre}, i.e. four balls with only one radius different from
zero. A natural question is what we can say about the classical case of four
hard balls (with all possible pairwise interactions allowed). For this system
even ergodicity has been shown in \cite{4golyo} if $\nu\ge 3$, thus we focus
our analysis on showing hyperbolicity for $\nu=2$.
As to the dynamical-topological part, the analogue of Lemma~\ref{LHdyn} has
been settled for any number of balls in any dimensions by a weak ball avoiding
Lemma which uses strongly the symmetries of hard ball systems (see
e.g. section 5 from \cite{ngolyo}). Thus in the rest of the appendix our aim
is to demonstrate how the proof of Proposition~\ref{PHalg} can be adapted to
four hard balls.
Our basic reference is section 4 from \cite{4golyo}. Rich collision sequences
for the system of four balls were classified there (up to equivalence) in
eleven cases. Cases 9-11 are analogous to cases (a) and (b) from
Proposition~\ref{PHalg} of this paper. Handling these sequences is much
simplified by the fact that they contain subsequences sufficient with respect
to a suitable sub-billiard dynamics. This is the strategy used in \cite{4golyo}
as well; we may repeat the argumentation word by word with the only difference
that because of $\nu=2$ we get $codim(N)\ge 1$ instead of $codim(N)\ge 2$ for
the submanifold of non-sufficient points.
In cases 1-8 however the proofs of \cite{4golyo} do not go through as they use
the dimensional restriction (see also Remark 4.28. in that paper). These
collision sequences are similar to our case (e) from Proposition~\ref{PHalg}
(actually case 1 is exactly our case (e)). Observe that for a cylindric
billiard equivalent to a hard ball system it is always true that either $L_a
\cap A_b =\{ 0 \}$ or $L_a\perp L_b$, thus we need not care about situations
analogous to subcase (e2) above. For brevity we have chosen one from cases
2-8, namely case 7, for which we show $codim(N)\ge 1$ by an adaptation of our
Lemma~\ref{LHalg}. All the other collision sequences are treated in a much
similar way.
In case 7 the collision sequence is $(\{ 3,4\} ,\{ 1,4\} ,\{ 1,2\} ,\{ 1,3\}
,\{ 1,4\})$ (here $\{ i,j\}$ means a collision of the two balls $i\ne j$). In
our notation we closely follow Lemma~\ref{LHalg}. Thus the advances are:\\
$\alpha'$ for the collision $\{ 3,4\}$;\\
$\alpha$ for the collision $\{ 1,3\}$;\\
$\beta$ for the first collision $\{ 1,4\}$ and $\beta'$ for the second.\\
Moreover we choose $\{ 1,2\}$ as our central collision with advance 0. The
non-collision time moments we fix are:\\
$t_*$ after $\{ 3,4\}$ and before the first $\{ 1,4\}$;\\
$t_-$ after the first $\{ 1,4\}$ and before $\{ 1,2\}$;\\
$t_+$ after $\{ 1,2\}$ and before $\{ 1,3\}$;\\
$t_{\#}$ after $\{ 1,3\}$ and before the second $\{ 1,4\}$.\\
The form of a velocity vector $v$ at any time moment is:
$$
v=(v_1,v_2,v_3,v_4)
$$
where, by the convention of zero total momentum we have $v_1+v_2+v_3+v_4=0$
(here $v_i$ means the two-dimensional velocity vector of the ball $i$). The
neutral vectors $w$, which we calculate at time moments $t_-$ and $t_+$ (by
zero advance this two neutral spaces coincide) have the same form.
Neutrality with respect to the central collision $\{ 1,2\}$ means
$w_1=w_2$. Moreover, by the form of the generator subspace for any collision
in a hard ball system and by the general formula (\ref{cpf}), we get the
following four equations for our neutral vector:
\begin{eqnarray}
\nonumber
w_3-w_4 &=& (\alpha'-\beta)(v_3^*-v_4^*) + \beta(v_3^--v_4^-);\\
\nonumber
w_1-w_4 &=& \beta(v_1^--v_4^-);\\
\label{4cpf}
\nonumber
w_1-w_3 &=& \alpha(v_1^+-v_3^+);\\
\nonumber
w_1-w_4 &=& (\beta'-\alpha)(v_1^{\#}-v_4^{\#}) + \alpha(v_1^+-v_4^+).\\
\end{eqnarray}
By the second and the fourth of the above equations we know that for
non-sufficient points:
\be
\exists\lambda\in {\bf R}; \lambda\ne 0: \quad \left( (v_1^--v_4^-)-\lambda (v_1^+-v_4^+)
\right) \;\|\; (v_1^{\#}-v_4^{\#})
\label{4fix}
\ee
while if we subtract the second equation in (\ref{4cpf}) from the third, then
together with the first equation we get that {\it for the same} $\lambda$:
\be
\left( (v_1^--v_3^-)-\lambda (v_1^+-v_3^+) \right) \;\|\; (v_3^*-v_4^*)
\label{4mozg}
\ee
We may handle the degenerate possibilities of zero advances (e.g. if
$\alpha=0$) or parallelity (e.g. if $(v_1^--v_4^-) \,\|\, (v_1^+-v_4^+)$)
exactly the same way as in Lemma~\ref{LHalg}. Otherwise apply a purely
configurational translation
$$
\delta q = (\delta q_1,\delta q_1,\delta q_1, -3\delta q_1)
$$
at time $t_-$. By the structure of the generator spaces neither of the
velocity vectors $v^-,v^+,v^*$ changes, thus if the point remains
nonsufficient, by (\ref{4fix}), the value $\lambda$ does not change throughout
the perturbation (remember parallelity is excluded). As a consequence the left
hand side of (\ref{4mozg}) remains constant, thus for the perturbed velocity
$\bar{v}^*$ the difference $\bar{v}^*_3-\bar{v}^*_4$ is parallel to the
original $v^*_3-v^*_4$. This component however moves on an arc of a circle.
\subsection*{Acknowledgements}
First of all I am much greatful to my supervisor, Domokos Sz\'asz for
suggesting me this topic and helping me a lot in all my considerations. Careful
reading of the manuscript and lots of useful remarks are thankfully
acknowledged for N\'andor Sim\'anyi and Imre P\'eter T\'oth.\\
This research was partially supported by Hungarian National Foundation for
Scientific Research, grant OTKA T26176.
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