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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 $a<0<b$ and $a$, $b$, $0$ are not moments of collision. \\ ${\cal N}_0
(S^{[a,b]}x)$, the
{\bf neutral subspace} at time $0$ for the segment $S^{[a,b]}x$ is defined as follows:
\begin{eqnarray*}
{\cal N}_0 (S^{[a,b]}x):=&\{& w\in {\bf R^d} : \exists(\delta>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 $\tau_j$;  $a<\tau_1< \cdots <\tau_n <b$ the moments of
collision on the segment, i.e. for any $j=1,...,n$ it is true that
$S^{\tau_j}x=(q_j,v_j) \in \partial M$. Let us assume $q_j \in \partial
(C_{l(j)})$, i.e. at time $\tau_j$ reflection occurs at the cylinder
$C_{l(j)}$. The finite sequence of symbols $(l(1), \cdots ,l(n))$ is
called the {\it symbolic collision sequence} of the trajectory segment. \\ 
A finite trajectory segment is said to be {\it connected} if it holds for the
corresponding symbolic collision sequence that the system of subspaces
$(A_{l(1)}, \cdots ,A_{l(n)})$ is transitive in the sense as it is formulated
at conjecture~\ref{Ctr}. \\ A finite trajectory segment is said to be {\it
rich} if the corresponding symbolic collision sequence contains 'enough
collisions' in some combinatorical sense. Formulating the concept of richness
so vaguely is done by purpose: the useful notion of richness depends and has
always depended on the specific model one considers. Furthermore, the useful
formulation also depends on what one wants to prove; for proving ergodicity
definitely a stronger notion of richness is needed than for showing
hyperbolicity. Nevertheless, the concepts of connectedness and richness should
be related in a way. With any notion of richness being used we call a phase
point rich/connected if it has a rich/connected finite trajectory segment and
we call it poor if it is not rich.\\
Dynamics of a phase point in a finite time interval is mainly characterized by
its symbolic collision sequence. It may happen, however, that trajectories
with different collision sequences show the same dynamical behaviour. For
example two consecutive collisions with the same cylinder have the same effect
as if only one collision occured. An {\it island} is a maximal subsequence of
a collision sequence consisting of consecutive collisions with the same
cylinder (see \cite{2heng,3golyo}). Collision sequences with the same island
structure may be treated as equivalent. We shall turn back to the question of
equivalence and to the correct formulation of richness at the end of
subsection~\ref{secAltalg}.

Following tradition (e.g. \cite{2heng,4golyo}) proving either hyperbolicity or
ergodicity is done in several steps. For the proof of hyperbolicity one needs
\\ (H1) {\it Geometric-algebraic considerations} to prove that $\mu$-a.e. rich
point in the phase space is sufficient; and \\ (H2) {\it Dynamical-topological
considerations} to prove that $\mu$-a.e. point of the phase space is rich.\\
We get sufficiency $\mu$-almost everywhere if (H1) and (H2) hold
simultaneously, this together with the above mentioned local hyperbolicity
theorem implies the hyperbolicity of the dynamics.

Ergodicity is a more difficult task as one has to ensure that the local
ergodic components indeed do make up one ergodic component (i.e. they are not
separated by codimension 1 subsets). It has to be shown that the 'exceptional
sets' are, besides being of zero $\mu$-measure, slim subsets of the phase
space (a subset is slim if it can be covered by a countable union of closed zero measure sets of topological
codimension 2, for a detailed analysis see the recent review \cite{gomb}). A
further difficulty is that one has to consider the singular orbits (they can be ignored in the proof of hyperbolicity as they
form a zero measure set (a set of codimension 1)). The steps of the proof of
ergodicity are:\\ (E1) {\it Geometric-algebraic considerations} to prove that
the set of points that are rich and non-sufficient is slim;\\  (E2) {\it
Dynamical-topological considerations} to prove that the set of poor (and,
simultaneously, non-sufficient) points is slim; and\\ (E3) Considerations on the
{\it singularities} of the system, including the verification of the
Chernov-Sinai Ansatz. \\ Putting parts (E1-E3) together implies, on the basis
of the local ergodicity theorem, the ergodicity of the billiard dynamics.

\subsection{Calculation of neutral subspaces}
\label{secAltalg}
    
In the study of various cylindrical billiards (especially hard ball systems,
see e.g. \cite{4golyo,ngolyo}) geometric-algebraic considerations are usually
much more difficult than the rest of the proof. This is true for our results
as well -- as the reader shall see from sections ~\ref{secErg} and ~\ref{secHyp},
while dynamical-topological considerations are more or less easy adaptations
of methods discussed in the literature, some new ideas are needed in the
geometric-algebraic part. 

Our task in the geometric-algebraic part is
approximately the following. Let us assume a trajectory segment with one
particular rich symbolic collision sequence is given (there is always a finite
number of symbolic collision sequences one has to study, depending on the
specific notion of richness we use for the model). If we knew how the neutral
subspace at a given (non-collision) time moment of the segment explicitly
looks like, we could conclude that the neutral subspace is one dimensional
(i.e. sufficiency holds) unless some degeneracy of the projections of velocity
vectors at different time moments occurs. Thus it is very important to find
explicit methods for the calculation of neutral subspaces of different
collision sequences.    

As to Hard Ball Systems, remarkable progress has
been achieved related to this problem through the so called {\it Connected
Path Formula} invented by N. Sim\'anyi (see \cite{nandi}). This formula
however, which has been so powerfully used since its invention (see
e.g. \cite{ngolyo,nandi}), uses the properties of hard ball
systems (the conservation of momenta), thus it cannot be applied
directly to
the wider class of general cylindric billiards. In this subsection some
general observations on the calculation of neutral subspaces are made which
will be of further use in the rest of the paper playing the role of the
connected path formula. Actually, when applied to the isomorphic cylindric
billiard in the case of particle systems with hard ball interactions, our
Lemma~\ref{Lcpf}, together with the conservation of momenta,
gives back the classical Connected Path Formula.  

From here on $S^{[a,b]}x$ is a fixed trajectory segment with collision
sequence $(l(1),...l(n))$; the moments of collision we denote by
$a<\tau_1(x)<...<\tau_n(x)<b$, while $t\in [a,b]$ is an arbitrary
non-collision time moment. $v(t)$ stands for the velocity vector at time $t$
and $w(t)$ for an arbitrary fixed vector of the neutral subspace, also
calculated at time $t$ (we use the natural isomorphism of neutral subspaces at
different time moments mentioned in subsection~\ref{secAltpr}). As before,
$P_K(w)$ denotes the orthogonal projection of the vector $w$ onto the subspace
$K$. \\
For all the collisions $i=1,...,n$ we define linear functionals $\alpha_i:
{\cal N}_t (S^{[a,b]}x) \mapsto {\bf R}$, the {\it advances} of the collisions
(see e.g. \cite{nandi}) as the unique linear extensions of the linear
functions $\alpha'_i$ defined in an open neighborhood of the origin of ${\cal N}_t
(S^{[a,b]}x)$: 
$$
\alpha'_i(w(t)):=\tau_i(x) - \tau_i(S^{-t}T_{w(t)}S^tx).
$$   
Here $T_{w(t)}(x)=T_{w(t)}(q,v)=(q+w(t),v)$ denotes the pure spatial
translation by the vector $w(t)$. By the natural isomorphism of neutral
subspaces the value of an advance does not depend on the time moment when the
neutral vector is calculated.

It is also easy to conclude (see e.g. \cite{2heng}) that for any $t$ such that
$\tau_{i-1}<t<\tau_i$ or $\tau_i<t<\tau_{i+1}$:  
\be 
w(t)=\alpha_i P_{L_{l(i)}}(v(t)) + a
\label{1heng}
\ee 
where $a\in A_{l(i)}$ is an element of the generator subspace which (just like
the value of the advance for the neutral vector $w$, $\alpha_i$) does
not depend on time. 

% and $\alpha$ is a real number, the so-called {\it advance
% of the collision}, also independent of time. More precisely, the advance is a
% real-valued linear functional defined on the neutral subspace, determined
% time-independently by the collision itself (see \cite{2heng}).\\
% Observe that the relation (\ref{1heng}) between the velocity vector and a
% neutral vector is true for times both before and after the collision with the
% cylinder $C_i$. It is also straightforward to see that if a vector is neutral
% at time $t$ for a trajectory segment, than it is neutral for any subsegment
% which contains the time moment $t$. Now we are in the position of stating our
% main Lemma for the calculation of neutral subspaces in cylindric billiards.

\begin{lemma}    
We use the notations introduced above, thus our collision sequence is \\
$(l(1),...,l(n))$, collision moments are denoted by $\tau_i$, the advances
by $\alpha_i, \; i=1,...,n$ (more precisely, the $\alpha_i$-s are the values of
the advances for a fixed neutral vector). Non-collision time moments $t_i$
are fixed between the consecutive collisions, $\tau_i<t_i<\tau_{i+1}$.
Our aim is to calculate the neutral subspace at time $t_{n-1}$, i.e directly
before the last collision. The following equations are valid for any
$j=1,...,(n-1)$ and any neutral vector $w(t_{n-1})$:  
\begin{eqnarray}
\nonumber
P_{L_{l(j)}}(w(t_{n-1}))= & & (\alpha_j-\alpha_{j+1})P_{L_{l(j)}}(v(t_j))
+ \; ... \; +\\ & & (\alpha_{n-2}-\alpha_{n-1})P_{L_{l(j)}}(v(t_{n-2}))+ \alpha_{n-1} P_{L_{l(j)}}(v(t_{n-1}))
\label{cpf}
\end{eqnarray}
N.B. For the projection of any such neutral vector onto the base subspace of the last cylinder in the collision sequence by (\ref{1heng}) it trivially holds: 
$$
P_{L_{l(n)}}(w(t_{n-1}))=\alpha_n P_{L_{l(n)}}(v(t_{n-1})) 
$$ 
\label{Lcpf} 
\end{lemma}

{\bf Proof.} For brevity we introduce the notation: $P_i$ for $P_{L_{l(i)}}$, $v^i$ for $v(t_i)$ and $w^i$ for $w(t_i)$. From (\ref{1heng}) the following series of equations are straightforward: 
$$ 
w^i-w^{i-1}=P_i(w^i)-P_i(w^{i-1})=\alpha_i P_i(v^i)-\alpha_i P_i(v^{i-1})  
$$ 
We only need one more well-known observation for the proof; the dynamics of a velocity vector is determined by specular reflections, thus while colliding with $C_i$ it's projection onto the generator subspace $A_i$ does not change: 
$$ 
P_i(v^i)-P_i(v^{i-1})=v^i-v^{i-1} 
$$ 
Putting the above two equations together and projecting orthogonally onto {\it any} $L_j, \; j=1,...,n$ we get: 
\be 
P_j(w^i)=P_j(w^{i-1}) + \alpha_i(P_j(v^i)-P_j(v^{i-1})) 
\label{telescope} 
\ee 
On the other hand from (\ref{1heng}) it trivially follows that  
$$ 
P_j(w^j)=\alpha_j P_j(v^j) 
$$ 
This together with (\ref{telescope}) summed over $i=(j+1) ... (n-1)$ gives the telescopic expression (\ref{cpf}). $\Box$ 
 
The advantage of the Lemma is that in case our collision sequence is long enough (i.e. $\Sigma_{i=1}^{n-1} L_{l(i)}={\bf R^d}$), any neutral vector is determined by the values of the advances through the simple equations (\ref{cpf}).   
  
The already mentioned equivalence of collision sequences is one more issue to be discussed briefly before closing this subsection (an analogous discussion has already appeared in \cite{4golyo}, principle 4.1) . We say that two collision sequences are {\it
 equivalent} if they can be transformed into each other by \\ 
(i) time direction change for the whole sequence, i.e. $(l(1),...,l(n)) \sim (l(n),...,l(1))$; \\     
(ii) doubling a collision with the same cylinder, i.e. $(..,l,l,..) \sim (..,l,..)$; \\ 
(iii) by the interchange of two consecutive collisions if the corresponding
cylinders have orthogonal base subspaces; i.e. $(..,l,m,..) \sim
(..,m,l,..)$ whenever $L_m \perp L_l$ holds. \\  
As it will be clear from the considerations in subsections ~\ref{secEalg} and
~\ref{secHalg}, these collision sequences are indeed equivalent from the viewpoint of geometric-algebraic considerations. The fact that the structure of neutral subspaces at
the corresponding time moments (i.e. in the cutting time moments, see below) is the same is a straightforward consequence of
the definitions and the equations above. The advantage of this notion is
that it is enough to consider only the shortest equivalent forms of all
possible rich collision sequences.\\
In the dynamical-topological part, however, one has to be more careful with
the equivalence of collision sequences. The following notion, suggested by
N. Sim\'anyi (\cite{archip}), helped a lot in a clearer formulation of the
dynamical-topological considerations. In addition to islands we introduce the
concept of {\it archipelago}. An archipelago is a maximal subsequence
$(l_1,...,l_n)$ of a symbolic collision sequence for which it is true that any two cylinders
present are either identical ($L_{l(i)}=L_{l(j)}$) or orthogonal
($L_{l(i)}\perp L_{l(j)}$). It is easy to see that (modulo time direction
change) two collision sequences are equivalent if and only if they have the
same archipelago structure. One more important notion is that of a {\it
cutting time moment} which we define as a non-collision time moment between
two consecutive archipelagos. Now we may turn back to the question of a
suitable formulation of richness, which should be invariant under equivalence
transformations. (It is straightforward to see that the concept of
connectedness depends solely on the archipelago structure). For proving
hyperbolicity double connectedness is a possibly good notion of richness in general; i.e. the trajectory segment is rich if there is a
cutting time moment $t$, $a<t<b$, such that both segments $S^{[a,t]}x$ and
$S^{[t,b]}x$ are connected. In fact, this is the notion of richness we use in
section~\ref{secHyp}.     
 

\section{Theorem~\ref{Te} -- pairwise transversal generator subspaces} 
\label{secErg}
\setcounter{equation}{0}
 
The subject of this section is a cylindric billiard with scatterers $C_1,...,C_k$ which, as it is formulated in Theorem~\ref{Te}, in addition to transitivity condition have the property $A_i\cap A_j=\{ 0\}$ whenever $i\ne j$. For  symbolic collision seque
nces we use the notation $\Sigma = (a,b,..)$, here the symbol $a$ denotes the cylinder $C_a$ with generator subspace $A_a$. Lowercase Latin letters $a,b,c,..$ denote the cylinders and the corresponding Greek letters $\alpha,\beta,\gamma..$ denote the adva
nces of the collisions, respectively. For vectors $w$ of the neutral subspace and the velocity vector $v$ upper indices refer to the time moments, while lower indices indicate orthogonal projections onto base subspaces. For instance $v_a^+$ is the orthogo
nal projection of the velocity vector at time $t_+$ onto the subspace $L_a$ (nevertheless the symbol $P_{L_a}(v(t_+))$ already introduced in the previous section may be used as well for the same quantity). The notation $K_1+K_2$ is used for the subspace o
f ${\bf R^d}$ spanned by to given (not necessarily orthogonal) subspaces $K_1$ and $K_2$. 

\subsection{Geometric-Algebraic Considerations}   
\label{secEalg}

\begin{proposition}
Assume $\Sigma=(a,b,c)$ is an arbitrary collision sequence of three collisions
that has no shorter equivalent form (the possibility $a=c$ is not excluded). For any regular phase point $x\in M^0$ a
bounded trajectory segment of which has collision sequence $\Sigma$ 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{P1kd}
\end{proposition}

{\bf Proof.} The strategy is the following. Let us assume we have a non-sufficient phase point $x$ fixed for which the collision sequence has the structure $(a,b,c)$ described above. We will show that $x$ lies on a one-codimensional submanifold $N$. The a
dvances (for a fixed neutral vector) are $\alpha, \beta$ and $\gamma$; furthermore arbitrary fixed non-collision time moments directly before and after collision with $b$ are denoted by $t_-$ and $t_+$, respectively. According to the possible geometrical 
position of the cylinders we distinguish three cases. 
 
{\bf Case 1.} The base subspaces for the first two collisions are transversal and non-orthogonal, thus $L_a\cap L_b=\{ 0\}$ and $L_a \not\perp L_b$. 
 
We examine a neutral vector $w$ at time moment $t_+$. By Lemma~\ref{Lcpf}: 
\be 
w_a=(\alpha -\beta )v_a^- + \beta v_a^+; \qquad w_b=\beta v_b^+ 
\label{proj} 
\ee 
Let us denote by $J$ the linear mapping of ${\bf R^d}$ onto itself which is evaluated as the sum of the two orthogonal projections onto the subspaces $L_a$ and $L_b$ for any vector $z$, i.e. $Jz=z_a+z_b$. The transversality of the corresponding generator 
subspaces $A_a \cap A_b=\{ 0 \}$ required for our model, which is equivalent to $L_a+L_b={\bf R^d}$, ensures that $J$ is indeed a linear bijection. Summing the two equations in (\ref{proj}) and applying $J^{-1}$ we get: 
$$ 
w=\beta v^+ +(\alpha-\beta) J^{-1}(v_a^-)
$$ 
On the other hand it is clear from (\ref{1heng}) that for any neutral vector directly before collision with $c$: $w-\gamma v^+ \in A_c$, thus 
$$ 
(\beta -\gamma)v^+ +(\alpha -\beta) J^{-1}(v^-_a) \in A_c 
$$ 
This equation however implies that unless two consecutive advances are equal (which is equivalent to sufficiency for our case!)  

\be 
\exists\lambda\in{\bf R}; \lambda\ne 0: \quad v^+ +\lambda J^{-1}(v_a^-) \in A_c
\label{E1c1} 
\ee 
Let us now apply a purely configurational infinitesimal translation $\delta q$
to the phase point of the trajectory segment at time $t_+$. Under the effect
of all such translations, while $v^+$ remains the same, the velocity at $t_-$
may change from $v^-$ to $\bar{v}^-$ (more precisely, the velocity difference
$v^--\bar{v}^-$ moves on the surface of a $dim(L_b)$ dimensional sphere
inside $L_b$ that goes through the origin). Assume that  the perturbed
trajectory is not sufficient either, i.e. (\ref{E1c1}) holds for $\bar{v}^-$
with some $\bar{\lambda} \in {\bf R}$. This implies:  
$$ 
J^{-1}(\lambda v^-_a -\bar{\lambda}\bar{v}^-_a) \in A_c 
$$   
Now observe that by $L_a\cap L_b=\{ 0\}$ it is only true for vectors inside $A_b$ that their images under $J$ lie in $L_a$. This together with the required transversality of generator subspaces ($A_b\cap A_c=\{ 0\}$) and the bijectivity of $J$ gives: 
\be 
\lambda v^-_a=\bar{\lambda}\bar{v}^-_a \quad \Rightarrow \quad v^-_a \| \bar{v}^-_a 
\label{E1c1a} 
\ee  

We should distinguish two further subcases.\\ 
(i) $L_a\cap A_b=\{0 \}$. Under the effect of all perturbations $\delta q$ the velocity difference projected onto $L_a$ (i.e. $v_a^--\bar{v}_a^-$) moves on a surface that definitely contains an arc of an ellipse. Thus we get a contradiction with (\ref{E1c1a}), $x$ lies on a submanifold $N$ of codimension at least 1.\\ 
(ii) $L_a\cap A_b\ne\{0 \}$. Then by dynamics $P_{A_b}(v^-)=P_{A_b}(v^+)$, thus, as $v^+$ remains unchanged 
\be 
P_{L_a\cap A_b}(v^-)=P_{L_a\cap A_b}(\bar{v}^-) 
\label{E1c1a1}
\ee 
Now we shall see $N=N^{(1)} \cup N^{(2)}$ with both $N^{(1)}$ and $N^{(2)}$
one-codimensional. For the points $x\in N^{(1)}$ let $P_{L_a\cap
A_b}(v^-)=P_{L_a\cap A_b}(v^+)=0$, this submanifold is clearly (at least)
one-codimensional. On the submanifold $N^{(2)}$ we assume that such a
restriction of the velocity is not present. Then by (\ref{E1c1a1}) we see that
(\ref{E1c1a}) can only hold if $\lambda=\bar{\lambda}$,
i.e. $v_a^-=\bar{v}_a^-$. Nevertheless $v_a^-$ can only remain unchanged under
the effect of all perturbations $\delta q$ if $L_a \perp L_b$, something we
excluded from case 1. As a perturbation moves away from $N^{(2)}$, it is
definitely of codimension at least one.\\  
Thus in case 1 it is indeed true that sufficiency may only fail locally on a submanifold $N$ for which $codim(N)\ge 1$.  
 
{\it Remarks.} Observe that our argument does not use any assumptions on the
geometrical position of the cylinder $C_c$, geometrical restriction is only
put on the cylinders of consecutive collisions $a$ and $b$, and not on $b$ and
$c$. E.g. the case $L_b \perp L_c$ is also covered. If $L_b\cap L_c\ne\{ 0\}$,
the situation is even better. In addition to the characterization of non-sufficient points described above, by (\ref{1heng})
it is straightforward to see that non-equality of the consecutive advances
$\beta$ and $\gamma$ (i.e. non-sufficiency) can only hold if  
\be
P_{L_b\cap L_c}(v^+)=0 
\label{E1c1b}
\ee   
Thus for the submanifold $N$ of non-sufficient points in fact $codim(N)\ge 2$ is valid, as we can find two transversal submanifolds $N_1$ and $N_2$ such that $N=N_1 \cap N_2$ holds and $codim(N_j)\ge 1$ (directions in configurational and velocity space, j
ust like two velocity restrictions in orthogonal subspaces are always transversal). 
 
{\bf Case 2.} The subspaces $L_a$ and $L_b$ are not transversal, i.e. $L_a\cap L_b \ne\{0 \}$.  
 
Repeating the argument in the remarks after case 1 we get $P_{L_a\cap L_b}(v^-)=0$ which gives for the submanifold of non-sufficient points $codim(N)\ge 1$ readily. It is however interesting to note that in many cases we automatically get $codim(N)\ge 2$:
\\ 
(i) if $L_b\cap L_c=\{ 0\}$ and $L_b\not\perp L_c$; we are exactly in the situation discussed in the remarks above, thus $codim(N)\ge 2$;\\ 
(ii) if $L_b\cap L_c\ne\{ 0\}$, we have $N=N_1 \cap N_2$, where $N_2$ is defined by (\ref{E1c1b}) and $P_{L_a\cap L_b}(v^-)=0$ is valid for the points of $N_1$. These two are transversal, as applying all possible purely configurational translations $\delta q$ at time $t_-$, $v^-$ does not change (so we remain on $N_1$). On the other hand $v^+_b$ moves on a surface of a sphere of full dimension in $L_b$, thus (\ref{E1c1b}) does not remain true for any such $\delta q$. We get $codim(N)\ge 2$.\\ 
(iii) for the case $L_b\perp L_c$ we are satisfied with $codim(N)\ge 1$. 
 
{\bf Case 3.} The two consecutive base subspaces are orthogonal, $L_a
\perp L_b$. Observe that, from the viewpoint of Proposition~\ref{P1kd}, there
is no need to consider this case separately, as by time direction change it is
equivalent to one of the cases 1 or 2 (orthogonality of $a$ and $b$ together
with $b$ and $c$ cannot occur, in that case we would have a shorter equivalent
form). Nevertheless it is worth to get the explicit form of $N$ as a condition
on the velocity at time $t_+$ to prepare the proof of Proposition~\ref{P2kd}.  
 
Observe that by the assumed $A_a\cap A_b=\{ 0\}$ the condition $L_a \perp L_b$
means $L_a=A_b$ and $L_b=A_a$. In case $L_b\cap L_c\ne\{ 0\}$ (if $L_a\cap
L_c\ne\{ 0\}$, by equivalence, similarly) we get $N$ explicitly as
(\ref{E1c1b}). If no such intersection is present, then the cylinders $C_b$
and $C_c$ are transversal in a strong sense, i.e. $L_b\cap L_c=\{ 0\}; \;
A_b\cap L_c=\{ 0\}$ (this is the geometric condition that was required from
the two cylinders in \cite{2heng}). It is easy to see that under such
geometrical conditions $dim(A_c)\ge 2$. From here on the velocity vector $v$ and
the neutral vectors $w$ are always understood in time $t_+$. By orthogonality
(\ref{cpf}) gives a very simple form for any neutral vector:  
$$ 
w_a=\alpha v_a; \quad w_b=\beta v_b 
$$ 
By a standard argument similar to the one used in case 1: 
\be 
\exists\lambda\in{\bf R}, \lambda\ne 0: \quad v_a +\lambda v_b \in A_c 
\label{E1c3} 
\ee 
The set of such velocity vectors is of codimension at least 1. Indeed, let us assume that (\ref{E1c3}) holds for $v$ and for any $\bar{v}$ which is parallel to $(v+\delta v)$ for some $\delta v \in A_c$ (we may assume $v \not\in A_c$, otherwise we get cod
imension 2 readily). Let the corresponding constants in (\ref{E1c3}) be $\lambda$ and $\bar{\lambda}$, respectively. The condition on both vectors implies:  
$$ 
(\bar{\lambda} -\lambda)v_b + (\bar{\lambda}-1) \delta v_b \in A_c 
$$ 
As we assumed $A_b\cap A_c=\{ 0\}$ there is definitely a possible choice of $\delta v\in A_c$ such that $\delta v_b$ and $v_b$ are lineary independent. This -- by the  strong transversality mentioned above -- gives $\lambda=\bar{\lambda}=1$, and looking a
t (\ref{E1c3}) we get $v\in A_c$ as a contradiction. 
 
Thus we have seen $codim(N) \ge 1$ in all the three cases, the Proposition is proven. $\Box$ 
 
It is time to fix the notion of richness we want to use for our model. We say
that a trajectory segment is {\bf rich} if its collision sequence contains \\
(i) either four consecutive archipelagos;\\
(ii) or three archipelagos at least one of which consists of more than one island.\\
It is easy to see that such rich sequences, in any of their shortest possible
equivalent forms, contain a subsequence of
at least four consecutive collisions $(a,b,c,d)$ (with double occurrence of
the same cylinder not excluded) such that $L_a\perp L_b$ and $L_c\perp L_d$ does not hold simultaneously.
Observe furthermore that double connectedness of the collision sequence
(already mentioned in subsection~\ref{secAltalg}) would mean for our model the
presence of four consecutive archipelagos; thus the notion of richness we use
is weaker.   
 
 
\begin{proposition} 
Assume $\Sigma=(a,b,c,d)$ is (an above described subsequence of) an arbitrary rich collision sequence. For any regular phase point $x\in M^0$ a bounded trajectory segment of which has collision sequence $\Sigma$ there exist an open neighborhood $x\in U_0$
 and a submanifold $N$ of $M$ such that (i) $codim(N)\ge 2$ and (ii) $\forall y \in U_0 \setminus N$ is sufficient. 
\label{P2kd} 
\end{proposition} 
 
{\bf Proof.} Now we shall think of $x$ as a non-sufficient phase point with one particular collision sequence $\Sigma$, and show that $x$ lies on the two-codimensional submanifold $N$. On our collision sequence $\Sigma=(a,b,c,d)$ we fix some
arbitrary non-collision time moments $t_-$ between collisions $a$ and $b$,
$t_0$ between $b$ and $c$ and $t_+$ between $c$ and $d$. As for the rest of
the notation we adopt the conventions of Proposition~\ref{P1kd}. Quantities
without upper indices are calculated at time $t_0$. As before, we have to
discuss several cases.
  
{\bf Case 1.} $L_b\cap L_c\ne\{ 0\}$. As we want to consider (suitable subsequences of) rich sequences 
$L_a\perp L_b$ and $L_d\perp L_c$ cannot hold simultaneously. We may assume
e.g. $L_b\not\perp L_a$. Then a quick reference to case 2 in the  proof of
Proposition~\ref{P1kd} shows that $codim(N)\ge 2$ is ensured even by the
subsegment $(a,b,c)$ of $\Sigma$.  

%We may assume that $L_a \perp L_b$ and
%$L_c \perp L_d$ does not hold simultaneously. Indeed, assume the contrary,
%thus $A_a=L_b$ and $A_d=L_c$. Then $A_a\cap A_d \ne \{ 0\}$ which is only
%possible (by the assumed pairwise transversality of generator subspaces) if
%$a=d$. Then, as collisions corresponding to orthogonal base subspaces are
%interchangeable, our sequence would have a shorter equivalent form $\Sigma\sim
%(b,a,c)$.  
 
{\bf Case 2.} $L_b\cap L_c=\{ 0\}$, $L_a\cap L_b=\{ 0\}$ and $L_c\cap
L_d=\{ 0\}$.  From Proposition~\ref{P1kd} applied to two subsegments containing
three collisions we know that $N=N_1\cap N_2$. All we have to prove is the
transversality of the (at least 1 codimensional) submanifolds $N_1$ and
$N_2$. As double orthogonality is not allowed there are two subcases: \\  
%(i) $L_a \perp L_b$ and $L_c \perp L_d$. According to case 3 from
%Proposition~\ref{P1kd} $N_1$ is defined by (\ref{E1c3}) and $N_2$ is given as
%a similar condition on the velocity at $t_0$:   
%\be 
%\exists\nu\in{\bf R}; \nu\ne 0: \quad v_d +\nu v_c \in A_b 
%\label{E2c2} 
%\ee 
%Let us apply the velocity perturbation $v \rightarrow \bar{v} \| (v+\delta v)$ where
%$\delta v\in A_c$ is now the velocity component $v_d$. Under the effect of such
%translations the velocity $\bar{v}$ moves on an arc of a circle which we get
%as the intersection of the velocity sphere and the plane spanned by the
%velocity components $v_d$ and $v_c$. It is easy to see that such  a
%perturation is tangential to $N_2$ ((\ref{E2c2}) holds for $\bar{v}$ with
%some $\bar{\nu}\in{\bf R}$) and transversal to $N_1$ (the argument of case 3
%from Proposition~\ref{P1kd} goes through) unless $v_c=0$ (this later case is
%of codimension 2 itself).\\ 
(i) $L_a \perp L_b$ and $L_c \not\perp L_d$ (or the other way round). $N_1$
is defined as a velocity condition in $t_0$ (by equation (\ref{E1c3})), as for
$N_2$ we have (by case 1 from Proposition~\ref{P1kd}) two possibilities. If
there exists a purely configurational translation transversal to $N_2$, the
required transversality is trivial. Otherwise the points of $N_2$ are defined
by the velocity restriction 
\be
(P_{L_d\cap A_c}(v)=)P_{L_d\cap A_c}(v^+)=0
\label{E2c2a}
\ee
By $L_c \not\perp L_d$ there definitely exists a vector $0\ne \delta v\in A_c$ such that
$\delta v\perp L_d\cap A_c$. Apply the
velocity perturbation $v \rightarrow \bar{v} \| (v+\delta v)$. Tangentiality to $N_2$
is obvious, thus we are done if we show transversality to $N_1$. An adaptation of the argument from Case 3 in Proposition~\ref{P1kd} clearly works unless $v_b$ and $\delta v_b$ are lineary dependent for any choice of $\delta v$. If such an accidental situ
ation comes about then the (generally at least two dimensional) velocity component $v_b$ should be restricted to a line determined by the geometrical position of our cylinders. This gives $codim(N)\ge 2$ together with (\ref{E2c2a}) above by the transversa
lity of $A_c$ and $A_a=L_b$.

(ii) Both $L_a \not\perp L_b$ and $L_c \not\perp L_d$; $N_1$ and $N_2$ are
described by case 1 from Proposition~\ref{P1kd}. If one of them is defined by
velocity restriction, while the other has transversal direction in
configurational space, their transversality is trivial. Assume both $N_1$ and
$N_2$ have transversal directions in configurational space. To prove their
transversality we only have to find a translation $\delta q$ in configuration
space which is tangential to $N_2$ and transversal to $N_1$. We are in a
situation analogous to the one discussed at the end of section 3 in
\cite{2heng}: the existence of such a perturbation is ensured by $A_b \cap A_c=\{ 0\}$, by $L_b \cap L_c=\{ 0\}$ (which is equivalent to $A_b +A_c ={\bf R^d}$), and by
the fact that any $\delta q \in A_c$ is tangential to $N_2$. However, in case
both $N_1$ and $N_2$ are defined by velocity restrictions, again the pairwise
transversality of generator spaces ($A_b \cap A_c=\{ 0\}$) ensures
$codim(N)\ge 2$. 

{\bf Case 3.} In addition to $L_b\cap L_c=\{ 0\}$ e.g. $L_c\cap L_d\ne\{ 0\}$. The
required $codim(N)\ge 2$ we get from the collision sequence $(b,c,d)$ readily
unless $L_b \perp L_c$. If such an orthogonality is present, we get $N=N_1\cap
N_2$. $N_1$ is defined, independently of the geometrical position of the
cylinders $C_a$ and $C_b$, as a condition on the velocity $v^-$ at time $t_-$
(see case 3 from Proposition~\ref{P1kd}). For the phase points belonging to
$N_2$ the condition $P_{L_c\cap L_d}(v^+)=0$ holds at time $t_+$. This later
condition can be surely spoilt by applying purely configurational translations
at time $t_-$. Under the effect of such translations, however, the velocity
$v^-$ does not change. Thus we have found $\delta q$ tangential to $N_1$ and
transversal to $N_2$, the required transversality of $N_1$ and $N_2$ is
proven.  
 
We have seen $codim(N) \ge 2$ in all possible cases, thus the Proposition is
proven. $\Box$     

\subsection{Dynamical-Topological Considerations and finishing the proof} 
\label{secEdyn}

Let us denote by $M_p^0$ the set of regular and poor points on our phase
space. The aim of the dynamical-topological part would be to prove that
$M_p^0$ is a slim subset; there may exist, however, some trivial 1-codimensional submanifolds of non-sufficient points. Therefore we should
restrict our considerations on a subset $M^\#\subset M$. Just like in \cite{ort} and
\cite{2heng}, we define the complement of $M^{\#}$, $M\setminus M^\#$ as the set of points which have
trajectories without any collision in at least one of the non-trivial
sub-billiards of our original billiard (as to the notion of a sub-billiard see subsection~\ref{secAltre}). Applying the considerations of
Appendix 2 from \cite{ort} to our model, we see that $M^\#$ has a finite number
of connected components. In the rest of the subsection we will prove that any
of these components belongs to one ergodic component. Then the proof of
Theorem~\ref{Te} is finished exactly the same way as it was done in Lemma
A.2.3 in \cite{ort} by connecting the components of $M^\#$ with bundles of
orbits of positive measure.

Let us first examine the set of poor phase points a bit
closer. We have the natural
partition $M_p^0\cap M^{\#}= M_1^0 \cup M_2^0\cup M_{3-}^0$ where the lower
index $i$ refers to the number of archipelagos of the collision sequence of
$x\in M_i^0$. We use $M_{3-}^0$ as not all collision sequences with three
archipelagos are poor (more precisely, for $x\in M_{3-}^0$ there are three
archipelagos, each containing only one island). However, it is useful to
further subdivide $M_2^0$ into $M_{2-}^0$ and $M_{2+}^0$, where the points
$x\in M_{2-}^0$ are characterized by the fact that both archipelagos contain
only one island. This partition is useful as we may treat the points of
$M_1^0\cup M_{2-}^0$ and $M_{2+}^0\cup M_{3-}^0$ separately. Indeed, observe
that for the later one the statement of Proposition~\ref{P1kd} holds, a fact
we will strongly use in the proof of Lemma~\ref{LEdyn} below.     

% For the collision sequence $\Sigma=(a,b,c,d)$ of a %phase point $x
%\in M_{4p}^0$ there are two orthogonal pairs; $L_b\perp %L_c$ is not possible
%as in that case we would have a shorter equivalent form. %Thus $L_a \perp L_b$
%and  $L_c \perp L_d$, which for our model means %$A_a=L_b$ and
%$A_d=L_c$. Then we may also assume $L_b\cap L_c =\{ %0\}$. Indeed, otherwise we
%would have $A_a\cap A_d \ne \{ 0\}$ which is only
%possible (by the assumed pairwise transversality of %generator subspaces) if
%$a=d$. Then, as collisions corresponding to orthogonal %base subspaces are
%interchangeable, our sequence would have a shorter %equivalent form $\Sigma\sim
%(b,a,c)$. Altogether for any phase point $x\in %M_{4p}^0$:
%\be
%L_a \perp L_b; \quad L_c \perp L_d; \quad L_b \not\perp %L_c; \quad L_b \cap L_c =\{ 0\} 
%\label{Ed}
%\ee  
 
We introduce the notation $M_{ns}$ for the set of non-sufficient points of our
billiard. The essence of the dynamical-topological part is the following
Proposition:

\begin{proposition}
The set of points $M_p^0\cap M_{ns}\cap M^\#$ is slim.
\label{PEdyn}
\end{proposition}

{\bf Proof} of this Proposition, closely following section 4 from \cite{2heng},
consists of several steps according to the partition of the 'poor' set described above.

The slimness of $M_1^0$ we get by a trivial adaptation of the classical ball
avoiding Theorem from \cite{wart} (as for a detailed description see
\cite{gomb} as well). Indeed, for a phase point $x\in M_1^0$ the entire
trajectory avoids at least one cylinder (otherwise there would be more
archipelagos). On the other hand dynamics for such a phase point is determined
by a non-trivial sub-billiard. This sub-billiard is, however, modulo almost
periodic motion, either dispersive or the product of two dispersive
billiards. In the former case classical ball avoiding, in the later a simple
generalization (cf. Lemma 4.6 from \cite{ort}) does the job.  

As for the points $x\in M_{2-}^0$ the island structure of the trajectory is $(a,b)$ with $L_a \not\perp L_b$. We can repeat the
proof of Lemma 4.1 from \cite{2heng} word by word; problem only arises at the
analogue of Sublemma 4.5 where the geometry of the cylinders is used. However,
it is possible to refer to Proposition 3.10 from \cite{trans} where we find
the needed generalization readily. Thus we have seen the slimness of $M_1^0\cup M_{2-}^0$. It is enough to show something weaker for the rest of the 'poor' set to finish the proof of the Proposition:

\begin{lemma}
The subsets $N_0=M_{3-}^0 \cap M_{ns}$ and $N_0'=M_{2+}^0 \cap M_{ns}$ are both
slim. 
\label{LEdyn}
\end{lemma}  

{\bf Proof} of this Lemma again relies on \cite{2heng} (it is an adaptation of
the proof of Lemma 4.2). We shall discuss the slimness of $N_0$ in detail and
indicate how the proof can be easily adapted to $N_0'$. We may denote the
island sequence of the examined point $x\in N_0$
by $(a,b,c)$ and fix some non-collision time moments, $0$ between islands $a$
and $b$ and $t_+$ between islands $b$ and $c$ (N.B. these are cutting time moments). As no orthogonality is present, either
(i) $L_b \not\perp L_c$ and $L_b\cap L_c=\{0 \}$ simultaneously or (ii)
$L_a\cap L_b\ne \{0 \}$ (otherwise reverse time 
direction). By Proposition~\ref{P1kd} we know that there is a small
open neighborhood $U$ of $x$ and a codimension-one submanifold $N$ such that $U\cap
N_0 \subset N$. The explicit form of $N$ is defined by cases 1 and 2 from
Proposition~\ref{P1kd} for the possibilities (i) and (ii) above,
respectively. We define the set \\ \\ 
$
F_-:= \{ x\in N:$ the negative semi-trajectory of $x$ has no proper collision
with $C_b\} 
$ \\ \\
As $U\cap N_0$ is a subset of $F_-$, to finish the proof it is enough to show
that the closed set $F_-$ contains no open disk $D$ in $N$. Assume the
contrary. Now -- following \cite{2heng} -- we have to find manifolds $\gamma(y)$ that go through the points $y\in D$ transversally to $N$ (and thus to
$D$). We always think of these manifolds in an infinitesimal sense, i.e. with inner radius smaller than a fixed small $\epsilon$. By their help the set 
$$
\bar{D}:=\cup_{y\in D} \gamma(y) 
$$
is constructed, the negative semi-trajectory of which -- although does not avoid the original
cylinder $C_b$ necessarily -- surely does avoid the modified cylinder we
obtain by shrinking $C_b$ by $\epsilon$. Just like in \cite{2heng}, a quick
reference to weak ball-avoiding (see \cite{wart,gomb}) gives $\mu(\bar{D})=0$,
a contradiction.

The construction of $\gamma(y)$ should be done, due to the fact that we have
various cases, in a way a bit different from the one in \cite{2heng}. From here on $y=(q,v)$ is an arbitrary fixed point in $D$. First we
define the so-called pseudo-unstable manifolds:  
$$
\gamma_0^u(y)=\{ z=(q',v') : v'=v, \; q'-q \in A_a, \| q'-q\| <
\epsilon/2 \}
$$
\\ 
$
\gamma_e^u(y)=\{ z: dist(y,z) < \epsilon/2; \; dist((S_a)^tz,(S_a)^ty) \rightarrow
0$ exponentially fast as $t\rightarrow -\infty\}
$ \\ \\ 
And then construct $\gamma(y)$ as:
$$
\gamma(y)= \cup_{z\in \gamma_0^u(y)} \gamma_e^u(z)= \cup_{z\in \gamma_e^u(y)} \gamma_0^u(z)
$$
Here the notation $(S_a)^t$ stands for the dynamics determined by the
sub-billiard of the original system the configuration space of which we get by
cutting out only the cylinder $C_a$ from ${\bf T^d}$. Observe that $\gamma_e^u(y)$
is the unstable manifold through $y$ for this sub-billiard, while $\gamma_0^u(y)$ is exactly the manifold that was used as $\gamma(y)$ in \cite{2heng}. 
By the explicit form of the submanifolds $N$ (cases 1 and 2 in
Proposition~\ref{P1kd}, respectively) we know that these manifolds $\gamma$
are indeed transversal to $N$. (More precisely, $\gamma_e^u$ is always transversal to $N$ in case (ii), while in case (i) -- again by the pairwise transversality of generator spaces -- either $\gamma_0^u$ or $\gamma_e^u$ does the job). It is also straight
forward that the set
$\bar{D}$ avoids the shrunk cylinder. 

As to the points of $N_0'$ we choose $0$ as the cutting time moment between
the two consecutive archipelagos. On the negative semi-trajectory there are
collisions with two cylinders $C_a$ and $C_b$ such that $L_a\perp L_b$. The
first collision after $0$ is with some $C_c$, where $L_c\not\perp L_a$ and
$L_c\not\perp L_b$. After that there may follow some further collisions but
that makes no difference for us. In our local analysis we again have $U\cap
N_0' \subset N$ where $N$ is now described by case 3 from
Proposition~\ref{P1kd}. We define $F_-$ as the set of points on $N$ which do
not collide with $C_c$ in the past. Submanifolds for which transversality to
$N$ is to be shown (they trivially do avoid the shrunk cylinder):\\ \\
$
\gamma(y)=\{ z: dist(y,z) < \epsilon; \; dist((S_{a,b})^tz,(S_{a,b})^ty) \rightarrow
0$ exponentially fast as $t\rightarrow -\infty\}
$ \\ \\
Here $(S_{a,b})^t$ is the sub-billiard dynamics we get by cutting out only the two orthogonal cylinders $C_a$ and $C_b$. The transversality of $\gamma(y)$ and $N$ is easily seen if $L_a\cap L_c \ne \{ 0\}$ or $L_b\cap L_c \ne \{ 0\}$, as in that case $N$ 
is described by (\ref{E1c1b}). If on the contrary no such intersection is present, the required transversality may be shown indirectly. Indeed, points of $N$ are described by the (at least) one-codimensional velocity restriction (\ref{E1c3}). Now it is ea
sy to see that if we combine $\gamma(y)$ with the infinitesimal perturbation (\ref{EDp}) below and project onto the velocity sphere, all points of ${\bf S^{d-1}}$ in a small open neighborhood of $v$ can be covered. However, by the explicit form of (\ref{E1c3}) the velocity translation 
\be
v=v_a +v_b \; \rightarrow \; \bar{v}= (1-\epsilon) v_a + (1+\epsilon) v_b 
\label{EDp}
\ee   
is clearly tangential to $N$.\\
The demonstration of our Lemma is
ready. $\Box$ Hence the Proposition is proven $\Box$.   
 
Our last task is to consider singular trajectories. More precisely, we should
prove the analogues of Propositions~\ref{P2kd} and ~\ref{PEdyn} for points of
$M^1$ on the one hand and verify the Chernov-Sinai Ansatz on the other
hand. Following tradition (\cite{4golyo,ort,2heng}) we state
Proposition~\ref{Psing} below from which both the Ansatz and the analogous
Lemmas follow. In our setting {\it eventually simple phase points} are points
of singular reflections $x\in {\cal SR}^+$ for which (i) the positive
semi-trajectory $S^{(0,\infty)}x$ is regular and (ii) there exists a time
moment $t>0$ 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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