\input amstex.tex \documentstyle{amsppt} \magnification=\magstep 1 \TagsOnRight \define\traj{S^{(-\infty,\infty)}x_0} \document {\catcode\@=11\gdef\logo@{}} \advance\baselineskip 6pt %\hsize 14truecm \noindent March 21, 1997 \bigskip \bigskip \centerline{Ergodicity of Hard Spheres in a Box} \vskip1cm \centerline{{N\'andor Sim\'anyi} \footnote{Research supported by the Hungarian National Foundation for Scientific Research, grants OTKA-7275 and OTKA-16425.}} \centerline{Bolyai Institute of J\'ozsef Attila University} \centerline{1 Aradi V\'ertanuk tere, Szeged, H-6720 Hungary} \centerline{E-mail: simanyi\@math.u-szeged.hu} \bigskip \bigskip \hbox{\centerline{\vbox{\hsize 8cm {Abstract.} We prove that the system of two hard balls in a $\nu$-dimensional ($\nu\ge 2$) rectangular box is ergodic and, therefore, actually it is a Bernoulli flow.}}} \bigskip \bigskip \centerline{1. Introduction} \bigskip \bigskip \heading The Model and the Theorem \endheading \bigskip Let us consider the billiard system of two hard balls with unit mass and radius $r$ ($02$ took fourteen years, and was done by Chernov and Sinai in [21]. Although the model of two hard balls in $\Bbb T^\nu$ is already rather involved technically, it is still a so called strictly dispersive billiard system, i. e. such that the smooth components of the boundary $\partial\bold Q$ of the configuration space are strictly convex from inside $\bold Q$. The billiard systems of more than two hard spheres in $\Bbb T^\nu$ are no longer strictly dispersive, but just dispersive (strict convexity of the smooth components of $\partial\bold Q$ is lost, merely convexity persists!), and this circumstance causes a lot of additional technical troubles in their study. In the series of my joint papers with A. Kr\'amli and D. Sz\'asz [9--12] we developed several new methods, and proved the ergodicity of more and more complicated semi-dispersive billiards culminating in the proof of the ergodicity of four billiard balls in the torus $\Bbb T^\nu$ ($\nu\ge 3$), [12]. Then, in 1992, Bunimovich, Liverani, Pellegrinotti and Sukhov [1] were able to prove the ergodicity for some systems with an arbirarily large number of hard balls. The shortcoming of their model, however, is that, on one hand, they restrict the types of all feasible ball-to-ball collisions, on the other hand, they introduce some extra scattering effect with the collisions at the strictly convex wall of the container. The only result with an arbirarily large number of spheres in a flat torus $\Bbb T^\nu$ was achieved in [15--16], where I managed to prove the ergodicity (actually, the K-mixing property) of $N$ hard balls in $\Bbb T^\nu$, provided that $N\le\nu$. The annoying shortcoming of that result is that the larger the number of balls $N$ is, larger and larger dimension $\nu$ of the ambient container is required by the method of the proof. Finally, in our latest joint venture with D. Sz\'asz [17] we prevailed over the difficulty caused by the low value of the dimension $\nu$ by developing a brand new algebraic approach for the study of hard ball systems. That result, however, only establishes hyperbolicity (nonzero Lyapunov exponents almost everywhere) for $N$ balls in $\Bbb T^\nu$. The ergodicity is a bit longer shot... None of the above results took up the problem of handling hard balls in physically more realistic containers, e. g. rectangular boxes. The extra technical hardship in their investigation is caused by the loss of the total momentum and center of mass. This amounts to the increase in the dimension of the configuration (phase) space without any additional scattering effect as a compensation. The problem of proving ergodicity for $N$ hard spheres in a $\nu$-dimensional rectangular box is so difficult that we were only able to achieve that goal in the case $N=2$. \bigskip \heading The Strategy of the Proof \endheading \bigskip After reviewing the necessary technical skills in Section 2, in the subsequent section we introduce the concept of combinatorial richness of the symbolic collision structure of a trajectory (segment), and then we prove that --- apart from some codimension-two, smooth submanifolds of the phase space --- such a combinatorial richness implies the sufficiency (hyperbolicity) of the trajectory. The proof of the theorem goes on by an induction on the number $k=0,1,\dots ,\nu$. Section 4 uses the inductive hypothesis (the statement of the theorem for $k-1$) and proves that the set of combinatorially non-rich phase points is {\it slim}, i. e. it can be covered by a countable family of closed, zero measure sets with codimension at least two. Section 5 contains the inductive proof of the theorem based upon the preceding two sections and the celebrated Theorem on Local Ergodicity for semi-dispersive billiards by Chernov and Sinai, [21]. The closing Appendix is a brief overview of a special orthogonal cylindric billiard which emerged in Section 4. \bigskip \bigskip \centerline{2. Prerequisites} \bigskip \bigskip \centerline{Semi-dispersive Billiards} \bigskip A billiard is a dynamical system describing the motion of a point particle in a connected, compact domain $\bold Q\subset{\Bbb R}^d$ or $\bold Q\subset{\Bbb T}^d=\Bbb R^d/\Bbb Z^d$, $d\ge 2$, with a piecewise $C^2$-smooth boundary. Inside $\bold Q$ the motion is uniform, whereas the reflection at the boundary $\partial\bold Q$ is elastic (the angle of reflection equals the angle of incidence). Since the absolute value of the velocity is a first integral of the motion, the phase space of our system can be identified with the unit tangent bundle over $\bold Q$. Namely, the configuration space is $\bold Q$, while the phase space is $\bold M=\bold Q\times\Bbb S^{d-1}$, where $\Bbb S^{d-1}$ is the unit $d-1$-sphere. In other words, every phase point $x$ is of the form $(q,v)$ where $q\in\bold Q$ and $v\in\Bbb S^{d-1}$ is a tangent vector at the footpoint $q$. The natural projections $\pi:\,\bold M\to\bold Q$ and $p:\,\bold M\to\Bbb S^{d-1}$ are defined by $\pi(q,v)=q$ and $p(q,v)=v$, respectively. Suppose that $\partial\bold Q=\cup_1^k\partial\bold Q_i$, where $\partial\bold Q_i$ are the smooth components of the boundary. Denote $\partial\bold M=\partial\bold Q\times\Bbb S^{d-1}$, and let $n(q)$ be the unit normal vector of the boundary component $\partial\bold Q_i$ at $q\in\partial\bold Q_i$ directed inwards $\bold Q$. The flow $\left\{S^t:\, t\in{\Bbb R}\right\}$ is determined for the subset $\bold M'\subset\bold M$ of phase points whose trajectories never cross the intersections of the smooth pieces of $\partial\bold Q$ and do not contain an infinite number of reflections in a finite time interval. If $\mu$ denotes the (normalized) Liouville measure on $\bold M$, i.e. $d\mu(q,v)=\text{const}\cdot dq\cdot dv$, where $dq$ and $dv$ are the differentials of the Lebesgue measures on $\bold Q$ and on $\Bbb S^{d-1}$, respectively, then under certain conditions $\mu(\bold M')=1$ and $\mu$ is invariant [8]. The interior of the phase space $\bold M$ can be endowed with the natural Riemannian metric. The dynamical system $\left(\bold M,\{S^t\},\mu\right)$ is called a {\it billiard}. Notice, that $\left(\bold M,\{S^t\},\mu\right)$ is neither everywhere defined nor smooth. The main object of the present paper is a particularly interesting class of billiards: the {\it semi-dispersive billiards} where, for every $q\in\partial\bold Q$ the second fundamental form $K(q)$ of the boundary is positive semi-definite. If, moreover, for every $q\in \partial\bold Q$ the second fundamental form $K(q)$ is positive definite, then the billiard is called a {\it dispersive billiard}. \medskip As it is pointed out in previous works on billiards, the dynamics can only be defined for trajectories where the moments of collisions do not accumulate in any finite time interval (cf. Condition 2.1 of [10]). An important consequence of Theorem 5.3 of [23] is that -- for semi-dispersing billiards -- there are {\it no trajectories at all with a finite accumulation point of collision moments}, see also [6]. \bigskip \centerline{Convex Orthogonal Manifolds} \bigskip In the construction of invariant manifolds a crucial role is played by the time evolution equation for the second fundamental form of codimension-one submanifolds in $\bold Q$ orthogonal to the velocity component $p(x)$ of a phase point $x$. Let $x=(q,v)\in\bold M\setminus\partial\bold M$, and consider a $C^2$-smooth, codimension-one submanifold $\tilde{\Cal O}\subset\bold Q\setminus\partial\bold Q$ such that $q\in\tilde{\Cal O}$ and $v=p(x)$ is a unit normal vector to $\tilde {\Cal O}$ at $q$. Denote by $\Cal O$ the normal section of the unit tangent bundle of $\bold Q$ restricted to $\tilde {\Cal O}$. (The manifold $\Cal O$ is uniqely defined by the orientation $(q,v)\in\Cal O$.) We call $\Cal O$ a {\it local orthogonal manifold} with support $\tilde\Cal O$. Recall that {\it the second fundamental form} $B_{\Cal O}(x)$ of $\Cal O$ (or $\tilde{\Cal O}$) at $x$ is defined through $$v(q+\delta q)-v(q)=B_{\Cal O}(x)\cdot\delta q+ o\left(\Vert\delta q\Vert\right),$$ and it is a self-adjoint operator acting in the $(d-1)$-dimensional tangent hyperplane of $\tilde{\Cal O}$ at $q$. \medskip A local orthogonal manifold $\Cal O$ is called {\it convex} if $B_{\Cal O}(y)\ge 0$ for every $y\in\Cal O$. Recall that the common tangent space $\Cal T_q\bold Q$ of the parallelizable configuration space $\bold Q\subset\Bbb R^d$ (or $\bold Q\subset\Bbb T^d$) at the point $q\in\text{int}\bold Q$ is the Euclidean space $\Bbb R^d$. \bigskip \centerline{Neutral Vectors, Advance and Sufficiency} \bigskip Consider a {\it non-singular} trajectory segment $S^{[a,b]}x$ in a semi-dispersive billiard. Suppose that $a$ and $b$ are {\it not moments of collision}. \medskip \proclaim{Definition 2.1} The neutral space $\Cal N_0(S^{[a,b]}x)$ of the trajectory segment $S^{[a,b]}x$ at time zero (a<00) \text{ such that }\forall\alpha\in(-\delta,\, \delta) \\ & p\left(S^a\left(\pi(x)+\alpha w,\, p(x)\right)\right)=p(S^ax)\text{ and} \\ & p\left(S^b\left(\pi(x)+\alpha w,\, p(x)\right)\right)=p(S^bx)\big\}. \endaligned $$\endproclaim \medskip It is known (see (3) in Section 3 of [21]) that \Cal N_0(S^{[a,b]}x) is a linear subspace of \Bbb R^d indeed, and V(x)\in\Cal N_0(S^{[a,b]}x). The neutral space \Cal N_t(S^{[a,b]}x) of the segment S^{[a,b]}x at time t\in [a,b] is defined as follows:$$ \Cal N_t(S^{[a,b]}x)=\Cal N_0\left(S^{[a-t,b-t]}(S^tx)\right). $$It is clear that the neutral space \Cal N_t(S^{[a,b]}x) can be canonically identified with \Cal N_0(S^{[a,b]}x) by the usual identification of the tangent spaces of \bold Q along the trajectory S^{(-\infty,\infty)}x (see, for instance, Section 2 of [10]). \medskip Our next definition is that of the {\it advance}. Consider a non-singular orbit segment S^{[a,b]}x with a certain collision \sigma taking place at time t=t(x,\, \sigma). For x=(q,v)\in\bold M and w\in\Bbb R^d, \Vert w\Vert sufficiently small, introduce the notation T_w(q,v):=(q+w,v). \medskip \proclaim{Definition 2.2} For any collision \sigma of S^{[a,b]}x and for any t\in[a,b], the advance$$ \alpha_\sigma :\, \Cal N_t(S^{[a,b]}x)\rightarrow\Bbb R $$is the unique linear extension of the linear functional defined in a sufficiently small neighborhood of the origin of \Cal N_t(S^{[a,b]}x) in the following way:$$ \alpha_\sigma (w):=t(x,\sigma)-t(S^{-t}T_w S^tx,\sigma). $$\endproclaim \medskip The overall important feature of the neutral space \Cal N_0(S^{[a,b]}x) is the following one: For every given ball-to-ball collision \sigma of the trajectory S^{[a,b]}x, for every fixed time t0)} \bigskip \bigskip In this section we will be studying the neutral spaces and sufficiency of {\it non-singular} trajectory segments \omega=S^{[a,b]}x_0, where a and b are not moments of collision. We are interested in answering the following fundamental question: How does the neutral space \Cal N(\omega) depend on the symbolic collision sequence \Sigma(\omega)=\Sigma of the given orbit segment \omega=S^{[a,b]}x_0, and what kind of combinatorial {\it richness} of \Sigma guarantees sufficiency, i. e. that \text{dim}\Cal N(\omega)=1? \bigskip \centerline{The Symbolic Sequence \Sigma(\omega)} \bigskip Let us symbolically denote the really dispersive (i. e. ball-to-ball) collisions \{1,2\} of the orbit segment \omega by \sigma_0,\sigma_1,\dots ,\sigma_n just as they follow each other in time, and denote by a0. Denote by U_0 the \epsilon_0-neighborhood of the base point x_0 in J_1:$$ U_0=\left\{x\in J_1|\, d(x_0,x)<\epsilon_0 \right\}. $$Besides \epsilon_0, select another, small positive number \delta_0 such that for every x\in U_0 the trajectory segment S^{[0,\delta_0]}x does not have a collision, not even a non-scattering ball-to-wall collision. We will foliate U_0 by {\it convex, local orthogonal manifolds} (see Section 2) as follows: We introduce the equivalence relation \sim in U_0 by the formula$$ x\sim y\Leftrightarrow (\exists \, \lambda\in\Bbb R) \quad q_1^0(x)-q_1^0(y)= q_2^0(y)-q_2^0(x)=\lambda\cdot e_{j_1} \tag 3.11 $$for x,y\in U_0, where e_{j_1} is the standard unit vector in the positive direction of the j_1-st coordinate axis. It is easy to see that, indeed, the equivalence classes C(x)\subset U_0 of \sim are (\nu+k-1)--dimensional, smooth, connected submanifolds of U_0. Furthermore, the positive images S^t\left(C(x)\right) (00 and j_1\in Z_l. Such an index l_0 exists by the assumption of the lemma. By using an induction on the integers l=0,1,\dots ,l_0-1, easy calculation proves that for every non-negative integer l0, and x_0 is also a smooth point of the corresponding singularity set \Cal S\ni x_0. We claim that \hat J_1 and \Cal S intersect each other transversally at x_0. \endproclaim \subheading{Proof} Following the proof of the previous lemma, we again consider the foliation U_0\cap \hat J_1=\bigcup_{\alpha\in A}C(x_\alpha), based on the equivalence relation (3.11), where A is a suitable index set. As we have seen in the previous proof, for every small t>0 the set S^t\left(C(x_\alpha)\right) is, indeed, a local, convex, orthogonal manifold. Sublemma 4.2 of [10] then yields that the manifold C(x_0)\ni x_0 is transversal to \Cal S at the base point x_0. Hence the lemma follows. \qed \bigskip \bigskip \centerline{4. Richness Is Abundant} \bigskip \bigskip In this section we will be investigating {\it non-singular} trajectories$$ S^{(-\infty,\infty)}x_0=\left\{x_t=(q_1^t,q_2^t;\, \dot q_1^t,\dot q_2^t) |\, t\in\Bbb R\right\} $$of the standard (\nu,k,r)-flow with k>0. (Sometimes we supress the time derivatives \dot q_i^t, and just simply write x_t=(q_1^t,q_2^t).) The singular orbits will be taken care of in Section 5. First of all, we note that, according to the theorems by Vaserstein [23] and Galperin [6], the moments of collisions on the trajectory S^{(-\infty,\infty)}x_0 can not accumulate at a finite point, i. e. every bounded time interval only contains finitely many collisions. The next thing we want to make sure is that there are infinitely many scattering (i. e. ball-to-ball'') collisions both in S^{[0,\infty)}x_0 and in S^{(-\infty,0]}x_0. \medskip \proclaim{Lemma 4.1} There exist finitely many nonzero vectors z_1,\dots ,z_l in \Bbb R^{\nu-k} with the following property: If a non-singular trajectory S^{(-\infty,\infty)}x_0 has no ball-to-ball collision on a time interval (t_0,\infty), then either (1) x_0 belongs to some slim exceptional subset of the phase space \bold M, \noindent or (2) the projection P_{\overline{\Cal A}}\left(v_1^{t_0}-v_2^{t_0} \right)\in\Bbb R^{\nu-k} of the relative velocity (measured at t_0) into the component \Bbb R^{\nu-k} of the velocity space \Bbb R^\nu=\Bbb R^k\oplus\Bbb R^{\nu-k}=\Bbb R^{\Cal A}\oplus \Bbb R^{\overline{\Cal A}} is perpendicular to some vector z_i. In either case the entire trajectory S^{(-\infty,\infty)}x_0 does not contain a single ball-to-ball collision. \endproclaim \medskip \subheading{Remark} In the case k=\nu we have that l=0 and the second possibility (2) will not occur. \medskip \subheading{Proof} Assume that t_0=0, i. e. the positive orbit$$ S^{[0,\infty)}x_0=\left\{x_t=(q_1^t,q_2^t)|\, t\ge 0\right\} $$does not have any ball-to-ball collision. Then there is a quite standard method of unfolding'' this positive orbit by reflecting the container \bold C=[0,1]^k\times\Bbb T^{\nu-k} across its boundary hyperplanes (q)_j=0, (q)_j=1, j=1,\dots ,k. Namely, we select first an arbitrary Euclidean lifting \tilde q_1^0,\, \tilde q_2^0\in\Bbb R^\nu of the initial positions q_1^0,\, q_2^0\in[0,1]^k\times\Bbb T^{\nu-k} by not altering the first k coordinates, that is, the non-periodic ones. Then we extend this lifting to t\ge 0 in a linear manner: \tilde q_i^t:=\tilde q_i^0+tv_i^0, t\ge 0, i=1,2. (We note that (\tilde q_1^t,\, \tilde q_2^t) is {\it not} a lifting of the original orbit.) Finally, we project the linear orbit \left\{(\tilde q_1^t,\, \tilde q_2^t)|\, t\ge 0 \right\} into the torus$$ \left(\Bbb R^k/2\cdot\Bbb Z^k\right)\times\Bbb T^{\nu-k}= \Bbb R^\nu/\left[\left(2\cdot\Bbb Z^k\right)\times\Bbb Z^{\nu-k}\right] $$by using the natural projection$$ \Bbb R^\nu\longrightarrow \Bbb R^\nu/\left[\left(2\cdot\Bbb Z^k\right)\times\Bbb Z^{\nu-k}\right], $$and obtain the positive trajectory \left\{(\hat q_1^t,\, \hat q_2^t)|\, t\ge 0\right\}. It is obvious that the finally obtained orbit is independent of the initial selection of the lifting. Let us define the canonical folding map''$$ \Phi=(\phi,\dots ,\phi;\, \text{id},\dots ,\text{id}):\, \left(\Bbb R^k/2\cdot\Bbb Z^k\right)\times\Bbb T^{\nu-k}\rightarrow [0,1]^k\times\Bbb T^{\nu-k}=\bold C $$in such a way that the same 2-periodic rooftop function \phi:\, \Bbb R/2\cdot\Bbb Z\rightarrow [0,1], \phi(x):=d(x,\, 2\cdot\Bbb Z), acts on the first k components, while the identity function acts on the remaining coordinates. One easily sees that \Phi\left((\hat q_1^t,\, \hat q_2^t)\right)=(q_1^t,\, q_2^t)=x_t. The fact that the original positive orbit S^{[0,\infty)}x_0 has no ball-to-ball collision immediately implies these assertions: (i) d(\hat q_1^t,\, \hat q_2^t)>2r for all t\ge 0; (ii) d\left(\hat q_1^t,\, R_{\Cal A}\hat q_2^t\right)>2r for all t\ge 0, \noindent where d(.,\, .) is the usual Euclidean distance in \left(\Bbb R^k/2\cdot\Bbb Z^k\right)\times\Bbb T^{\nu-k} inherited from \Bbb R^\nu and the operator R_{\Cal A} multiplies the first k coordinates by minus one, see also the previous section. According to Lemma A.2.2 of [22], there are finitely many nonzero vectors w_1,\dots ,w_p\in\Bbb R^\nu (none of which is a scalar multiple of another) --- not depending on the phase point x_0 --- such that \langle v_1^0-v_2^0;\, w_{j_1}\rangle=0 for some j_1 (because of (i)), and \langle v_1^0-R_{\Cal A}v_2^0;\, w_{j_2}\rangle=0 for some j_2 (because of (ii)). If at least one of the two projections P_{\Cal A}w_{j_1} and P_{\Cal A}w_{j_2} is nonzero, then, as it follows from the independence of the vectors P_{\Cal A}(v_1^0-v_2^0) and P_{\Cal A}(v_1^0-R_{\Cal A}v_2^0)=P_{\Cal A}(v_1^0+v_2^0), the equations \langle v_1^0-v_2^0;\, w_{j_1}\rangle=0 and \langle v_1^0-R_{\Cal A}v_2^0;\, w_{j_2}\rangle=0 are independent, so they together a define a manifold with codimension two. Such phase points x_0 are listed up in part (1) of the lemma. Therefore, we can assume that w_{j_1} and w_{j_2} both belong to the second factor \Bbb R^{\nu-k} of the velocity space \Bbb R^\nu=\Bbb R^k\oplus\Bbb R^{\nu-k}. If w_{j_1}\ne w_{j_2} (i. e. they are not even parallel), then the equations \langle v_1^0-v_2^0;\, w_{j_1}\rangle=0 and \langle v_1^0-R_{\Cal A}v_2^0;\, w_{j_2}\rangle=0 are again independent, and such phase points x_0 are listed up in (1). The only way of obtaining a codimension-one fanily of exceptional phase points x_0 is then to have w_{j_1}=w_{j_2} and P_{\Cal A}w_{j_1}=0. List up the vectors w_j with P_{\Cal A}w_{j}=0 in a sequence z_1,\dots ,z_l (\in\Bbb R^{\nu-k}), and obtain the first statement of the lemma. The fact that for an exceptional phase point x_0 -- listed up in (1)-(2) -- the entire trajectory S^{(-\infty,\infty)}x_0 does not contain a single ball-to-ball collision immediately follows from the following, simple observation: \medskip \proclaim{Sublemma 4.2} Consider the two rays L^+,\, L^-\subset\Bbb R^n as follows:$$ L^+=\left\{q_0+tv_0|\, t\ge 0\right\},  L^-=\left\{q_0+tv_0|\, t\le 0\right\}, $$q_0\in\Bbb R^n, 0\ne v_0\in\Bbb R^n. Then d(L^+,\, \Bbb Z^n)=d(L^-,\, \Bbb Z^n). \endproclaim \subheading{Proof} Apply the natural projection \pi:\, \Bbb R^n\rightarrow \Bbb R^n/\Bbb Z^n=\Bbb T^n to the half lines L^{\pm}. It is well known that \text{Cl}\left(\pi L^+\right)=\text{Cl}\left(\pi L^-\right)=S is a coset with respect to a subtorus of \Bbb T^n. (Here \text{Cl} denotes the closure.) This means that if d(L^+,\, \Bbb Z^n)<\alpha, then S intersects the open ball with radius \alpha centered at zero in \Bbb T^n and, therefore, d(L^-,\, \Bbb Z^n)<\alpha, as well. This finishes the proof of Sublemma 4.2 and, hence, the proof of Lemma 4.1, as well. \qed \medskip \subheading{Remark} Part (2) of the assertion of 4.1 is really not vague. Indeed, in the case of a trajectory without a ball-to-ball collision the time evolutions of the q-components are {\it independent} of each other, thanks to the orthogonality of the walls of the container. Therefore, if the \overline{\Cal A}-component P_{\overline{\Cal A}}(v_1^0-v_2^0) of the initial relative velocity belongs to some exceptional hyperplane of \Bbb R^{\nu-k} (just as in the proof of Lemma A.2.2 of [22]) and the the initial relative position P_{\overline{\Cal A}}(q_1^0-q_2^0) belongs to some non-empty, open region, then even the \overline{\Cal A}-parts of the positions q_i^t (i=1,2) never get closer to each other than 2r, and there will be no ball-to-ball collision. \medskip Thus, by dropping the exceptional phase points x_0 listed up in (1) and (2) of 4.1 (the latter set of phase points will be discussed in the next section), we can assume that the non-singular trajectory \traj contains infinitely many ball-to-ball collisions in each time direction. Let us list all such collisions of \traj as (\dots,\sigma_{-1},\sigma_0,\sigma_1,\dots), so that \sigma_0 is the first collision occuring in positive time (t=0 is supposed not to be a moment of collision), i. e. there is not even a non-scattering, wall-to-ball collision in the time interval [0,t_0). Just as in the previous section, t_n=t(\sigma_n) denotes the time of the n-th collision, n\in\Bbb Z. Following (3.2), we can now speak about the sets Z_i(1), Z_i(2), Z_i\subset\Cal A, (i\in\Bbb Z). The doubly infinite sequence$$ \Sigma=\left(\dots,\, \sigma_{-1},\, Z_0,\, \sigma_0,\, Z_1,\, \sigma_1,\, \dots\right) $$is now called the symbolic collision sequence of the trajectory \omega=\traj. \medskip {\bf From now on, in this section we will always be assuming the following inductive hypothesis:} \medskip \proclaim{Hypothesis 4.3} For every integer k'2r for every t\in\Bbb R, while d\left(\hat q_1^{t_n(x)}(x),\, R_{\{1\}}\hat q_2^{t_n(x)}(x)\right)=2r (n\in\Bbb Z), where t_n(x)=t\left(\sigma_n(x)\right) is the time of the n-th scattering collision \sigma_n on the orbit of x\in U_0\cap P_1. The above discussed \hat{\bold C}-dynamics \left\{(\hat q_1^t(x),\, \hat q_2^t(x))|\, t\in\Bbb R\right\} (which is a standard (\nu,k-1,r)-flow without the normalizations (v_1)_1+(v_2)_1=(q_1)_1+(q_2)_1=0) can now be defined for every phase point x\in U_0 by using the initial lifting (\hat q_1^0(x),\, \hat q_2^0(x))=(q_1^0(x),\, q_2^0(x)) of the positions, irrespectively of whether x\in P_1 or not. (In other words, when defining this \hat{\bold C}-dynamics \left\{(\hat q_1^t(x),\, \hat q_2^t(x))|\, t\in\Bbb R\right\}, we do not remove the antipodal cylinder d\left(\hat q_1,\, R_{\{1\}}\hat q_2\right)<2r from the configuration space but, rather, we allow the above inequalities.) We choose a small number \epsilon_0>0 and define the following closed subsets of U_0 (see also (5.2) and (5.7) of [11]):$$ \aligned F_+=\left\{x\in U_0\big|\, d\left(\hat q_1^t(x),\, R_{\{1\}}\hat q_2^t(x) \right)\ge 2r \quad \forall \, t\ge 0\right\}, \\ F_-=\left\{x\in U_0\big|\, d\left(\hat q_1^t(x),\, R_{\{1\}}\hat q_2^t(x) \right)\ge 2r \quad \forall \, t\le 0\right\}, \endaligned \tag 4.7  \aligned F'_+=\left\{x\in U_0\big|\, d\left(\hat q_1^t(x),\, R_{\{1\}}\hat q_2^t(x) \right)\ge 2r-\epsilon_0 \quad \forall \, t\ge 0\right\}, \\ F'_-=\left\{x\in U_0\big|\, d\left(\hat q_1^t(x),\, R_{\{1\}}\hat q_2^t(x) \right)\ge 2r-\epsilon_0 \quad \forall \, t\le 0\right\}, \endaligned \tag 4.8 $$where, in the case of singular trajectories, we understand these inequalities in such a way that they should hold for {\it some} trajectory branch. This convention makes the sets F_{\pm}\subset U_0, F'_{\pm}\subset U_0 closed, just as in Main Lemma 5.1 of [11] or in (5.4)--(5.7) of [12]. It is obvious that$$ F_+\subset F'_+, \qquad F_-\subset F'_-, \tag 4.9 $$and$$ U_0\cap P_1\subset F_-\cap F_+. \tag 4.10 $$Since the \hat{\bold C}-flow \left\{(\hat q_1^t(x),\, \hat q_2^t(x))|\, t\in\Bbb R\right\} has the quantities (v_1)_1+(v_2)_1 and (q_1)_1+(q_2)_1 as first integrals, it is now quite natural to define a foliation$$ \aligned U_0 &=\cup\left\{H_{a,b}|\quad a\in I_1,\; b\in I_2 \right\} \\ H_{a,b} &:=\left\{x\in U_0|\, (q_1(x))_1+(q_2(x))_1=a,\; (v_1(x))_1+(v_2(x))_1=b \right\}, \endaligned \tag 4.11 $$where I_1,\, I_2\subset\Bbb R are suitable open intervals. (We can assume that the shape of the small, open neighborhood U_0\ni x_0 is such that it permits us to establish the union (4.11) with open intervals I_1, I_2.) It is an important consequence of the assumption 4.3/A that for \mu-almost every phase point x\in U_0 there exist the maximum--dimensional (actually, (\nu+k-2)-dimensional) local, exponentially stable and unstable invariant manifolds \gamma^s(x), \gamma^u(x) (containing x as an interior point) with respect to the \hat{\bold C}-dynamics, defineable as follows:$$ \aligned \gamma^s(x)=\text{CC}_x\left\{y\in U_0\big|\, d(\hat x^t,\hat y^t)\rightarrow 0 \text{ exp. fast as } t\rightarrow +\infty\right\}, \\ \gamma^u(x)=\text{CC}_x\left\{y\in U_0\big|\, d(\hat x^t,\hat y^t)\rightarrow 0 \text{ exp. fast as } t\rightarrow -\infty\right\}, \endaligned \tag 4.12 $$where$$ \hat x^t=(\hat q_1^t(x),\hat q_2^t(x);\, \hat v_1^t(x),\hat v_2^t(x)), $$and the \hat{\bold C}-phase point \hat y^t is defined analogously. (Here the symbol \text{CC}_x(.) denotes the connected component of a set containing the point x.) Clearly, \gamma^s(x)\cup\gamma^u(x)\subset H_{a,b} for x\in H_{a,b}. According to the fundamental integrability'' of slimness (see Section 2 or Lemma 3.8 of [15]), it is enough to show that the closed subset F_-\cap F_+\cap H_{a,b} of H_{a,b} is slim in H_{a,b} for every a\in I_1, b\in I_2. It follows immediately from the ergodicity assumption of 4.3/A that$$ \mu_{H_{a,b}}\left((F'_-\cup F'_+)\cap H_{a,b}\right)=0. \tag 4.13 $$Recall that F'_{\pm}\supset F_{\pm}. In order to show that the closed, zero set F_-\cap F_+\cap H_{a,b} has at least two codimensions in H_{a,b}, it is enough to prove that this set enjoys the non-separating property, see (ii) of Lemma 3.9 in [15]. This is what we will do. Besides the exponentially stable and unstable manifolds \gamma^s(x),\, \gamma^u(x)\subset H_{a,b} (x\in H_{a,b}), we need to use the one-dimensional neutral manifolds \gamma^0(x)\subset H_{a,b} corresponding to the \hat{\bold C}-flow direction. We set$$ \aligned \gamma^0(x)=\big\{y\in H_{a,b}\big|\, v_i(y)=v_i(x)\text{ and }\exists \lambda\in\Bbb R \text{ such that} \\ q_i(y)=q_i(x)+\lambda v_i(x)- \frac{\lambda b}{2}(1,0,\dots ,0) \quad (i=1,2)\big\} \endaligned \tag 4.14 $$for x\in H_{a,b}, a\in I_1, b\in I_2. Subtracting the vector \dfrac{\lambda b}{2}(1,0,\dots ,0) from the positions is just needed in order to project back into H_{a,b}. It is clear that the following transversality condition holds, see also (5.19) and 5.20 of [12]:$$ \Cal T_x H_{a,b}=\Cal T_x\gamma^u(x)+\Cal T_x\gamma^s(x)+\Cal T_x\gamma^0(x), \tag 4.15 $$where \Cal T_x(.) enotes the tangent space of a manifold at the foot point x and + is a notation for the direct sum (not necessarily orthogonal) of linear spaces. It is easy to see that for every x\in U_0, y\in\gamma^s(x) or y\in\gamma^0(x)$$ d\left(\hat q^t(x),\, \hat q^t(y)\right)\le d\left(\hat q^0(x),\, \hat q^0(y)\right)\le\text{diam}U_0 \text{ for } t\ge 0, \tag 4.16 $$and, analogously, for any pair x,y\in U_0 for which y\in\gamma^u(x) or y\in\gamma^0(x)$$ d\left(\hat q^t(x),\, \hat q^t(y)\right)\le d\left(\hat q^0(x),\, \hat q^0(y)\right)\le\text{diam}U_0 \text{ for } t\le 0. \tag 4.17 $$Therefore, if the size \text{diam}U_0 is chosen small enough compared to \epsilon_0, then we have the analogue of Lemma 5.8 from [11]: \medskip \proclaim{Sublemma 4.18} For any x\in U_0, if \gamma^s(x)\cap F_+\ne\emptyset (\gamma^0(x)\cap F_+\ne\emptyset), then \gamma^s(x)\subset F'_+ (\gamma^0(x)\subset F'_+). Analogously, if \gamma^u(x)\cap F_-\ne\emptyset (\gamma^0(x)\cap F_-\ne\emptyset), then \gamma^u(x)\subset F'_- (\gamma^0(x)\subset F'_-). \endproclaim \medskip According to our assumption 4.3, if the fixed base point x_0 does not belong to some exceptional, slim subset of \bold M determined by the \hat{\bold C}-dynamics, then x_0 is sufficient with respect to this \hat{\bold C}-flow and the fundamental statement of Lemma 3 of [21] (or Theorem 3.6 of [10]) holds true. (Since their formalism is too technical, we do not even quote them here.) As a direct corollary of these results, the absolute continuity of the triple of foliations \gamma^s(.), \gamma^u(.), \gamma^0(.) (see Theorem 4.1 of [7]), and the transversality relation (4.15), we obtain the crucial Zig-zag Lemma, the precise analogue of Corollary 3.10 of [10]: \medskip \proclaim{Sublemma 4.19 (Zig-zag Lemma)} Suppose that the base point x_0\in\bold M is sufficient with respect to the \hat{\bold C}-dynamics. Then we can select the open neighborhood U_0\ni x_0 so small that the following assertion holds true: \medskip For every manifold H_{a,b}\subset U_0 (a\in I_1, b\in I_2), for every open, connected subset G\subset H_{a,b}, and for almost every pair of points x,\, y\in G\setminus(F'_-\cup F'_+) one can find a finite sequence$$ \xi_1^u,\; \xi_1^0,\; \xi_1^s,\; \xi_2^u,\; \xi_2^0,\; \xi_2^s,\dots ,\xi_n^u,\; \xi_n^0,\; \xi_n^s $$of submanifolds of G with the following properties: \medskip (i) \xi_i^\alpha is an open, connected subset of \gamma^\alpha(z) for some z\in\xi_i^\alpha, \alpha=u,\, 0,\, s, i=1,\dots ,n; \medskip (ii) x\in\xi_1^u, y\in\xi_n^s; \medskip (iii)$$ \emptyset\ne\xi_i^u\cap\xi_i^0\subset G\setminus(F'_-\cup F'_+),\; \emptyset\ne\xi_i^0\cap\xi_i^s\subset G\setminus(F'_-\cup F'_+),\; \emptyset\ne\xi_i^s\cap\xi_{i+1}^u\subset G\setminus(F'_-\cup F'_+) $$for i=1,\dots ,n. \endproclaim \medskip \subheading{Remark} The non-empty intersections in (iii) must be one-point-sets, according to the transversality (4.15). \medskip It follows immediately from (i)--(iii) and Sublemma 4.18 that$$ \bigcup\left\{\xi_i^\alpha \big|\, \alpha=u,\, 0,\, s;\quad i=1,\dots ,n \right\}\subset G\setminus(F_-\cap F_+). \tag 4.20 $$Thus we obtained that almost every pair of points x,\, y\in G\setminus(F'_-\cup F'_+) can be connected by a continuous curve \left\{\xi(t)|\, 0\le t\le 1\right\} so that \xi(t)\in G\setminus(F_-\cap F_+) for 0\le t\le 1. Since the open subset G\setminus(F'_-\cup F'_+) has full measure in G (and, therefore, it is dense in G), we get that the open subset G\setminus(F_-\cap F_+) of G must be connected. This proves (ii) of Lemma 3.9 of [15], i. e. \text{dim}(F_-\cap F_+\cap H_{a,b})\le\text{dim}H_{a,b}-2 and, therefore, F_-\cap F_+ is indeed a slim subset of U_0. The proof of Lemma 4.5 is now complete. \qed \medskip The last lemma of this section takes care of the case k=\nu, |Z_i|\left(\nu-|Z_i|\right)=0, (i\in\Bbb Z). \medskip \proclaim{Lemma 4.21} Assume that k=\nu and |Z_i|\left(\nu-|Z_i|\right)=0 for every integer i. Then the phase point x_0 necessarily belongs to some slim, exceptional subset of \bold M. \endproclaim \medskip \subheading{Proof (A sketch)} Since the proof of this lemma is very similar to that of Lemma 4.5, we are not going to present it here in whole detail but, instead, we will just point out the differences between the two proofs. We again begin with unfolding'' the trajectory$$ \traj=\left\{\left(q_1^t(x_0),\, q_2^t(x_0)\right)\big|\; t\in\Bbb R\right\} \tag 4.22 $$by reflecting the cubic container \bold C=[0,1]^\nu across its faces (q)_j=0 and (q)_j=1, j=1,\dots ,\nu, pretty much the same way as we did in the proof of 4.5. Namely, we consider the covering map$$ \Phi=(\phi,\dots ,\phi):\; \hat{\bold C}=\Bbb R^\nu/2\cdot\Bbb Z^\nu \rightarrow [0,1]^\nu=\bold C \tag 4.23 $$with the rooftop map \phi:\, \Bbb R/2\cdot\Bbb Z\rightarrow[0,1], \phi(y):=d(y,\, 2\cdot\Bbb Z) acting on all components, see also (4.6). Then we just pull back the trajectory of (4.22) by the mapping \Phi to obtain the unfolded trajectory$$ \hat\omega(x_0)=\left\{\left(\hat q_1^t(x_0),\, \hat q_2^t(x_0)\right) \big|\; t\in\Bbb R\right\}, $$\hat q_i^t(x_0)\in\Bbb R^\nu/2\cdot\Bbb Z^\nu by using the selected initial pull-back$$ \left(\hat q_1^0(x_0),\, \hat q_2^0(x_0)\right)= \left(q_1^0(x_0),\, q_2^0(x_0)\right), \quad \text{mod }2\cdot\Bbb Z^\nu. $$The hypothesis |Z_i|\left(\nu-|Z_i|\right)=0 (i\in\Bbb Z) implies that the obtained unfolded curve$$ \hat\omega\subset\left(\Bbb R^\nu/2\cdot\Bbb Z^\nu\right)\times \left(\Bbb R^\nu/2\cdot\Bbb Z^\nu\right)=\hat{\bold C}\times\hat{\bold C} $$has reflections only at the cylinder$$ C=\left\{(\hat q_1,\, \hat q_2)|\; d(\hat q_1,\, \hat q_2)=2r\right\} $$and the antipodal'' cylinder$$ \overline{C}= \left\{(\hat q_1,\, \hat q_2)|\; d(\hat q_1,\, -\hat q_2)=2r\right\}. $$We can now define the unfolded orbit$$ \hat\omega(x)=\left\{\left(\hat q_1^t(x),\, \hat q_2^t(x)\right) \big|\; t\in\Bbb R\right\} $$for every phase point x=\left(q_1^0(x),\, q_2^0(x)\right)= \left(\hat q_1^0(x),\, \hat q_2^0(x)\right)\in U_0 by forgetting about the collisions at d\left(\hat q_1,\, R_Z\hat q_2\right)=2r with 0<|Z|<\nu (making these cylinders transparent), and retaining only the collisions at the cylinders C and \overline{C}. In the Appendix we show that the so obtained \hat{\bold C}-dynamics (\hat{\bold C}=\Bbb R^\nu/2\cdot\Bbb Z^\nu) with the configuration space \hat{\bold C}\times\hat{\bold C}\setminus(C\cup\overline{C}) is, in fact, a splitting orthogonal cylindric billiard in the sense of the article [22]. There is now one additional first integral, namely the energy \Vert \hat v_1+\hat v_2\Vert^2. Therefore, the foliation of the small open neighborhood U_0\ni x_0 (the analogue of (4.11)) is now going on by specifying the value of \Vert \hat v_1^0+\hat v_2^0\Vert^2=\Vert v_1^0+v_2^0\Vert^2:$$ \aligned U_0 &=\bigcup\Sb a\in I\endSb H_a, \\ H_a &:=\left\{x\in U_0 \big|\; \Vert v_1^0(x)+v_2^0(x)\Vert^2=a \right\}. \endaligned \tag 4.24 $$It is shown in the Appendix that, for any fixed value of \Vert\hat v_1+\hat v_2\Vert^2, the \hat{\bold C}-flow is the Cartesian product of two dispersive Sinai--billiards and, therefore, it is Bernoulli, and all but two of its Lyapunov exponents are nonzero. Thus, in this situation we have \text{dim}\gamma^s(x)=\text{dim}\gamma^u(x)=2\nu-2, \text{dim}\gamma^0(x)=2, \text{dim}H_a=4\nu-2, and for every x\in U_0 the linear direct sum of the tangent spaces of the transversal \gamma^s(x), \gamma^u(x) and \gamma^0(x) is exactly the tangent space of the folium H_a\ni x through x, see also (4.15). If the original orbit \omega(x)=S^{(-\infty,\infty)}x of a phase point x\in U_0 has the property |Z_i(x)|\left(\nu-|Z_i(x)|\right)=0 (i\in\Bbb Z), then this fact is reflected by the \hat{\bold C}-orbit \hat\omega(x) in such a way that the configuration point \left(\hat q_1^t(x),\, \hat q_2^t(x)\right) never enters a forbidden'' open region, namely the interiors of the cylinders d\left(\hat q_1,\, R_Z\hat q_2\right)\le 2r, 0<|Z|<\nu. This makes it possible to define the closed sets F_+, F_- just the same way as in the proof of 4.5. Also, by shrinking the forbidden region mentioned above, we can define the larger closed sets F'_+, F'_-. Thanks to the ergodicity of the \hat{\bold C}-flow, we again have$$ \mu_{H_a}\left((F'_-\cup F'_+)\cap H_a\right)=0. Plainly, the analogues of sublemmas 4.18--4.19 remain valid in the recent situation. By putting together these ingredients, we see that the closed set F_-\cap F_+\cap H_a has, indeed, at least two codimensions in H_a. Finally, by using again the integrability of slimness (Lemma 3.8 of [15]), we obtain that the points x\in U_0 with the property |Z_i(x)|\left(\nu-|Z_i(x)|\right)=0 (i\in\Bbb Z) form a slim subset of U_0. This finishes the proof of Lemma 4.21 and, therefore, the proof of Key Lemma 4.4, as well. \qed \bigskip \bigskip \centerline{5. Proof of the Theorem} \bigskip \bigskip By using the results of the preceding sections, here we prove our \medskip \proclaim{Theorem} The standard (\nu,k,r)-flow (\nu\ge 2, 0\le k\le\nu, 0n \right\}. \endaligned \tag 5.2 (If the positive orbitS^{[0,\infty)}x$is singular, then we understand the above requirements in such a way that they should hold for {\bf some} branch of$S^{[0,\infty)}x$, thus making the sets$F_+^{(1)}$,$F_+^{(2)}$closed in$\hat J_1$.) We claim that both sets$F_+^{(1)}$,$F_+^{(2)}$have an empty interior in$\hat J_1$. \endproclaim \medskip \subheading{Remark 5.3} In the case$k<\nu$the set$F_+^{(2)}$is contained in$F_+^{(1)}$(meaning$\cup_{l>n} Z_l(x)=\emptyset$for every$x\in F_+^{(2)}$), and the statement about$F_+^{(2)}$follows from the claim about$F_+^{(1)}$. The result$\text{int}_{\hat J_1} F_+^{(2)}=\emptyset$is only significant in the case$k=\nu$. \medskip \subheading{Proof} Since the proof is very similar to the proof of Lemma 6.1 of [15], here we will only present a rough sketch of it. Furthermore, the proof of$\text{int}_{\hat J_1} F_+^{(2)}=\emptyset$(in the case$k=\nu$) is quite analogous to the proof of$\text{int}_{\hat J_1} F_+^{(1)}=\emptyset$and, therefore, we will only be dealing with the result$\text{int}_{\hat J_1} F_+^{(1)}=\emptyset$. (This analogy is similar to the analogy between the proofs of lemmas 4.5 and 4.21.) Without restricting the generality, we assume that$j_2=1$. We argue by contradiction, so we suppose that$\text{int}_{\hat J_1} F_+^{(1)}\ne\emptyset$and, by choosing a smaller open piece of the manifold$\hat J_1$, we can assume that$F_+^{(1)}=\hat J_1$, i. e. the positive orbit$S^{[0,\infty)}x$of every point$x\in\hat J_1$follows a pretty irregular pattern:$1\notin\cup_{l=n+1}^\infty Z_l(x)$. Select now a base point$x_0\in\hat J_1$and a small, open ball neighborhood$U_0\subset\bold M(\nu,k,r)$of$x_0$. We can assume that the time point$b>0$is chosen in such a way that there is no collision (not even a wall collision) between$t_n(x)=t\left(\sigma_n(x)\right)$and$b$for every$x\in U_0$. Let us push forward the open set$U_0$and$U_0\cap\hat J_1$by the$b$-th power of the$(\nu,k,r)$-flow:$S^b(U_0)=W_0$,$S^b\left(U_0\cap\hat J_1\right)=K_0$,$S^b(x_0)=y_0$. By our assumption, all trajectories$S^{(0,\infty)}y$follow the irregular pattern $$1\notin\bigcup_{l=0}^\infty Z_l(y), \quad y\in K_0. \tag 5.4$$ Therefore, for every$y\in K_0$we can again unfold'' the positive orbit $$S^{(0,\infty)}y=\left\{\left(q_1^t(y),\, q_2^t(y)\right)|\, t\ge 0\right\}$$ by reflecting our container$\bold C=[0,1]^k\times\Bbb T^{\nu-k}$across two of its faces$(q)_1=0$,$(q)_1=1$, and obtain the positive orbit $$\hat{S}^{(0,\infty)}y=\left\{\left(\hat q_1^t(y),\, \hat q_2^t(y)\right)|\; t\ge 0\right\}$$ in the container $$\hat{\bold C}=\left(\Bbb R/2\cdot\Bbb Z\right)\times [0,1]^{k-1}\times \Bbb T^{\nu-k},$$$\hat q_i^t(y)\in\hat{\bold C}$, see also the proof of Lemma 4.5. Quite similarly to the proof of Lemma 4.5, the above defined$\hat{\bold C}$-dynamics $$\hat\omega(y)=\left\{\left(\hat q_1^t(y),\, \hat q_2^t(y)\right)|\; t\ge 0\right\} \quad (y\in K_0),$$ which is essentially a standard$(\nu,k-1,r)$-flow without the normalization$(v_1)_1+(v_2)_1=(q_1)_1+(q_2)_1=0$, can now be defined for every phase point$y\in W_0$by using the initial lifting$\left(\hat q_1^0(y),\, \hat q_2^0(y)\right)=\left(q_1^0(y),\,q_2^0(y)\right)$of the positions, irrespectively of whether$1\in\cup_{l=0}^\infty Z_l(y)$, or not. Consider now the codimension-two submanifold$\tilde W_0\subset W_0\aligned \tilde W_0= &\big\{y\in W_0\big|\;\left(q_1(y)\right)_1+\left(q_2(y)\right)_1= \left(q_1(y_0)\right)_1+\left(q_2(y_0)\right)_1 \\ &\text{and } \left(v_1(y)\right)_1+\left(v_2(y)\right)_1= \left(v_1(y_0)\right)_1+\left(v_2(y_0)\right)_1 \big\}, \endaligned \tag 5.5 and the intersection\tilde K_0=K_0\cap\tilde W_0$, corresponding to fixing the values of$(q_1)_1+(q_2)_1$and$(v_1)_1+(v_2)_1$according to what these values are for the base point$y_0=S^bx_0\in K_0$. The phase point$y\in\tilde W_0$and its positive$\hat{\bold C}$-orbit$\hat\omega(y)$can be naturally identified with the phase point and positive orbit of a standard$(\nu,k-1,r)$-flow by changing the reference coordinate system, i. e. by reducing the first component of$q_1+q_2$and$v_1+v_2$to zero. Although we will be using this equivalence, but, for the sake of brevity, we are not going to introduce an extra notation for that purpose. The meaning of (5.4) is that there exists a small, open ball$B(z_0,\epsilon_0)\subset\bold M(\nu,k-1,r)$of the phase space of the$(\nu,k-1,r)$-flow, such that for every$y\in\tilde K_0$the positive$\hat{\bold C}$-orbit $$\hat\omega(y)=\left\{\hat y^t=\left(\hat q_1^t(y),\, \hat q_2^t(y)\right)|\; t\ge 0\right\}$$ avoids the ball$B(z_0,\epsilon_0)$. We can now define the exponentially stable, local invariant manifolds$\gamma^s(x)\subset\tilde W_0$for almost every phase point$x\in\tilde W_0$, similarly to (4.12), as follows: $$\gamma^s(x)=\text{CC}_x\left\{y\in\tilde W_0\big|\, d(\hat x^t,\hat y^t) \rightarrow 0\text{ exp. fast as }t\rightarrow +\infty\right\}.$$ \medskip \proclaim{Sublemma 5.6} The codimension-one submanifold$\tilde K_0$of$\tilde W_0$is transversal to the invariant manifold$\gamma^s(y)\subset\tilde W_0$for every$y\in\tilde K_0$, whenever$\gamma^s(y)$is a manifold containing$y$as an interior point. \endproclaim \medskip \subheading{Proof} This transversality immediately follows from our combinatorial assumption ($j_1\notin Z_0(x)$or$j_1\in\cup_{l=1}^n Z_l(x)$for every$x\in U_0$) by the method of the proof of Lemma 3.10. We note that this is just the point of the proof of 5.1 where we use the above mentioned combinatorial hypothesis. \qed It follows now from the transversality sublemma and from the Transversal Fundamental Theorem for semi-dispersive billiards (Theorem 3.6 of [10]) that for almost every phase point$y\in\tilde K_0$(with respect to any smooth measure on$y\in\tilde K_0$) the set$\gamma^s(y)$is a submanifold of$\tilde W_0$containing$y$as an interior point. Therefore, the union $$E:=\bigcup\Sb y\in\tilde K_0\endSb \gamma^s(y)\subset\tilde W_0$$ has positive measure in$\tilde W_0$. However, if the size of the open set$W_0$is chosen small enough compared to the radius$\epsilon_0$of the avoided ball$B(z_0,\epsilon_0)$, then the already proved ball avoiding$\hat\omega(y)\cap B(z_0,\epsilon_0)=\emptyset$implies that the positive$\hat{\bold C}$-orbit$\hat\omega(y')$of any$y'\in\gamma^s(y)$avoids the shrunk ball$B(z_0,\epsilon_0/2)$, see also (4.16) and Sublemma 4.18. Thus, we obtained that in the$(\nu,k-1,r)$-dynamics the positive trajectory of every phase point$y\in E$avoids the open ball$B(z_0,\epsilon_0/2)$, and, yet$E$has positive measure in the phase space$\bold M(\nu,k-1,r)$. This contradicts our inductive hypothesis 4.3 postulating the ergodicity of the$(\nu,k-1,r)$-flow. Hence Lemma 5.1 follows. \qed \medskip \proclaim{Corollary 5.7} Assume the inductive Hypothesis 4.3. Then there exists a slim subset$\hat S\subset\bold M=\bold M(\nu,k,r)$of the phase space of the standard$(\nu,k,r)$-flow such that every phase point$x\in\bold M\setminus\hat S$either belongs to the union$\Cal F$of the codimension-one submanifolds featuring in part (2) of Lemma 4.1, or the trajectory$S^{(-\infty,\infty)}x$contains at most one singularity and it is {\bf sufficient}. Furthermore, the Chernov--Sinai Ansatz (see Section 2) holds true for the standard$(\nu,k,r)$-flow. \endproclaim \medskip \subheading{Proof} Lemma 4.1 says that for every phase point$x\in\bold M\setminus(\hat S_1\cup\Cal F)$($\hat S_1,\hat S_2,\dots$will always denote some slim subsets of$\bold M$) the orbit$S^{(-\infty,\infty)}x$contains infinitely many scattering (ball-to-ball) collisions in each time direction. Furthermore, thanks to Lemma 4.1 of [10], for every phase point$x\in\bold M\setminus(\hat S_2\cup\Cal F)$($\hat S_2\supset\hat S_1$) the trajectory$S^{(-\infty,\infty)}x$contains at most one singularity. Then key lemmas 4.4, 3.5 and lemmas 5.1, 3.10 imply that every non-singular phase point$x\in\bold M\setminus(\hat S_3\cup\Cal F)$($\hat S_3\supset\hat S_2$) has a sufficient orbit$S^{(-\infty,\infty)}x$. As far as the singular phase points$x\in\Cal S\Cal R^+$are concerned, Lemma 6.1 of [15] asserts that for a generic (with respect to the surface measure of$\Cal S\Cal R^+$) singular phase point$x\in\Cal S\Cal R^+$the positive orbit$S^{(0,\infty)}x$is non-singular and sufficient. This proves that \medskip (i) for every phase point$x\in\bold M\setminus(\hat S_4\cup\Cal F)$($\hat S_4\supset\hat S_3$) the orbit$S^{(-\infty,\infty)}x$contains at most one singularity, and it is sufficient; \medskip (ii) the Chernov--Sinai Ansatz holds true. This proves Corollary 5.7. \qed \medskip \proclaim{Corollary 5.8} Assume Hypothesis 4.3. Denote by$\bold M\setminus\Cal F=\bigcup_{i=1}^r \Omega_i$the decomposition of the open set$\bold M\setminus\Cal F$into its connected components, where $$\Cal F=\left\{(q_1,q_2;\, v_1,v_2)\in\bold M \big|\, P_{\overline{\Cal A}}(v_1-v_2)\perp z_j \text{ for some } j\le l \right\},$$ see part (2) of Lemma 4.1. We claim that every component$\Omega_i$($i=1,\dots ,r$) belongs to one ergodic component of the standard$(\nu,k,r)$-flow. \endproclaim \medskip \subheading{Remark} We note that in the case$k=\nu$the set$\Cal F$is empty ($l=0$) and, therefore,$r=1$. \medskip \subheading{Proof} Since the complement set$\Omega_i\setminus\hat S_4$is connected and it has full measure in$\Omega_i$, the statement immediately follows from the previous corollary and the Chernov--Sinai Theorem on Local Ergodicity, i. e. Theorem 5 of [21]. \qed \medskip The last outstanding task in the inductive proof of our Theorem is to connect the open components$\Omega_1,\dots ,\Omega_r$by (positive beams of) trajectories. This will be done by the method from the closing part of the proof of Lemma 4 of [9]. Namely, it is enough to construct for every vector$z_j\in\Bbb R^{\nu-k}$a piece of trajectory connecting a phase point$x^{(1)}=\left(q_1^{(1)},q_2^{(1)};\, v_1^{(1)},v_2^{(1)}\right)$with the property$\langle v_1^{(1)}-v_2^{(1)};\, z_j\rangle>0$by another phase point$x^{(2)}=\left(q_1^{(2)},q_2^{(2)};\, v_1^{(2)},v_2^{(2)}\right)$for which$\langle v_1^{(2)}-v_2^{(2)};\, z_j\rangle<0$. This can be done, however, very easily by slightly perturbing a tangential$(1,2)$-collision for which the normal vector of impact is parallel with the vector$z_j$and, therefore, the relative velocity of the tangentially colliding'' balls is perpendicular to$z_j$. Plainly, we can perturb this tangential collision in such a way that the perturbed collision is no longer singular, and the normal vector of impact does not change. In this way we obtain the required phase points$x^{(1)}$and$x^{(2)}$together with a piece of trajectory connecting them. This completes the inductive proof of our theorem. \qed \bigskip \bigskip \centerline{APPENDIX} \bigskip \centerline{A Special Orthogonal Cylindric Billiard} \bigskip \bigskip In the proof of Lemma 4.21 we unfolded the non-singular orbit of (4.22) (with$|Z_i|(\nu-|Z_i|)=0$for$i=0,\pm 1,\pm 2,\dots$) and thus obtained the other trajectory $$\hat\omega(x_0)=\left\{\left(\hat q_1^t(x_0),\, \hat q_2^t(x_0)\right)|\; t\in\Bbb R\right\},$$ with$\hat q_i^t(x_0)\in\Bbb R^\nu/2\cdot\Bbb Z^\nu=\Bbb T_2^\nu$,$i=1,2$. The so constructed orbit$\hat\omega(x_0)$has collisions only at the cylinder $$C=\left\{\left(\hat q_1,\, \hat q_2\right)\in\Bbb T_2^\nu \times \Bbb T_2^\nu\big|\; d\left(\hat q_1,\, \hat q_2\right)=2r \right\}$$ and at the antipodal'' cylinder $$\overline{C}= \left\{\left(\hat q_1,\, \hat q_2\right)\in\Bbb T_2^\nu \times \Bbb T_2^\nu\big|\; d\left(\hat q_1,\, -\hat q_2\right)=2r \right\}.$$ We need to understand the ergodic properties of the arising billiard flow in the configuration space$\bold Q=\Bbb T_2^\nu \times\Bbb T_2^\nu\setminus(C\cup\overline{C})$. For the sake of simplifying the notations, we will work with the unit torus$\Bbb T^\nu=\Bbb R^\nu/\Bbb Z^\nu$, instead of$\Bbb T_2^\nu$, and we will supress$\hat{\;}$from over$q$and$v=\dot q$. It turns out that the$\nu$-dimensional generator (constituent) spaces $$\Bbb T_+=\left\{(x,\, x)|\; x\in\Bbb T^\nu\right\}, \quad \Bbb T_-=\left\{(y,\, -y)|\; y\in\Bbb T^\nu\right\}$$ of the cylinders$C$and$\overline{C}$are obviously orthogonal. Thus, our system is an {\it orthogonal cylindric billiard} in the sense of [22]. Let us write the configuration point $$(q_1,\, q_2)\in\Bbb T^\nu\times\Bbb T^\nu\setminus\left(C\cup \overline{C}\right)$$ in the form$(q_1,\, q_2)=(x,\, x)+(y,\, -y)$. Since the group homomorphism$\Psi:\, \Bbb T_+\times\Bbb T_-\rightarrow \Bbb T^\nu\times\Bbb T^\nu$,$\Psi\left((x,x),\, (y,-y)\right)=(x+y,\, x-y)$is surjective with the kernel$\text{Ker}\Psi=\left\{\left((x,x),\, (x,x)\right)|\, 2x=0\right\}$, we obtain a$2^\nu$-to-$1$covering of the original flow if we switch from$(q_1,\, q_2)\in\Bbb T^\nu\times\Bbb T^\nu$to$\left((x,x),\, (y,-y)\right)$with$q_1=x+y$,$q_2=x-y$. In terms of$x$and$y$the conditions$d(q_1,q_2)\ge 2r$and$d(q_1,-q_2)\ge 2r$precisely mean that$d(y,G)\ge r$and$d(x,G)\ge r$, where$G:=\left\{g\in\Bbb T^\nu|\, 2g=0\right\}$. There are now two first integrals:$E_1=\dfrac{1}{2}\Vert\dot x\Vert^2=\dfrac{1}{8}\Vert\dot q_1+\dot q_2\Vert^2$and$E_2=\dfrac{1}{2}\Vert\dot y\Vert^2=\dfrac{1}{8}\Vert\dot q_1- \dot q_2\Vert^2$. Thanks to the orthogonality, the$(x,\, \dot x)$and$(y,\, \dot y)$parts of the covering dynamics evolve independently of each other. This means that --- after fixing the values of$\Vert\dot q_1+\dot q_2\Vert^2$and$\Vert\dot q_1-\dot q_2\Vert^2$--- the$2^\nu$-to-$1$covering flow is the product of two,$\nu$-dimensional, dispersive Sinai billiards. Therefore, the original orthogonal billiard flow is mixing and all but two of its Lyapunov exponents are nonzero. Besides that, for almost every phase point$z=(q_1,q_2;\, \dot q_1,\dot q_2)$of this orthogonal cylindric billiard (with fixed values of$\Vert\dot q_1+\dot q_2\Vert^2$and$\Vert\dot q_1-\dot q_2\Vert^2$, of course) the$(2\nu-2)$-dimensional, exponentially contracting and expanding manifolds$\gamma^s(z)$and$\gamma^u(z)$exist, and they contain$z$as an interior point. \bigskip \bigskip \subheading{Acknowledgement} A significant part of this work was done while the author enjoyed the warm hospitality and inspiring research atmosphere at the Department of Mathematics of The Pennsylvania State University, University Park Campus. \bigskip \bigskip \Refs \bigskip \bigskip \widestnumber\key{23} \ref\key 1 \by L. Bunimovich, C. Liverani, A. Pellegrinotti, Yu. Sukhov \paper Special Systems of Hard Balls that Are Ergodic \jour Commun. Math. Phys. \vol 146 \pages 357-396 (1992) \endref \medskip \ref\key 2 \by L. A. Bunimovich, Ya. G. 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