\input amstex.tex \documentstyle{amsppt} \magnification=\magstep 1 \TagsAsMath \define\sdb{semi-dispersing billiard} \define\sdbs{semi-dispersing billiards} \define\bil{billiard} \define\sinb{\hbox{[Sin(1970)]\;}} \define\sci{(\Sigma,C_i)} \def\bt{{\Bbb T}} \def\br{{\Bbb R}} \def\ba{{\Bbb A}} \def\bz{{\Bbb Z}} \def\boq{{\bold {Q}}} \def\bn{{\Bbb N}} \define\coK{K_{(\bar q_i,I_i,E_i)}} \define\coKt{K_{(\bar q_i +tI_i,I_i,E_i)}} \def\ks{\Cal S} \def\kr{\Cal R} \define\sr{\ks \kr ^+} \def\g{\gamma} \define\spix{S^{(0,\infty)}x} \def\bp{\Bbb P} \def\kt{\Cal T} \def\kv{\Cal V} \def\ke{\Cal E} \def\kn{\Cal N} \define\clc{C\ell_C (C_{ed}(t_0,P))} \define\ctp{C_{ed}(t_0,P)} \define\xu{U(x_0)} \define\yu{U(y_0)} \define\fie{\uph _{I,E}\,} \define\pik{\pi_{2,3,4}\,} \define\kssa{\hbox{[K-S-Sz (1989)]\;}} \define\kssb{\hbox{[K-S-Sz (1990)]\;}} \define\kssc{\hbox{[K-S-Sz (1991)]\;}} \define\kssd{\hbox{[K-S-Sz (1992)]\;}} \define\sima{\hbox{[Sim(1992)-I]\;}} \define\simb{\hbox{[Sim(1992)-II]\;}} \define\szaa{\hbox{[Sz (1995)]\;}} \define\szab{\hbox{[Sz (1994)]\;}} \define\ssz{\hbox{[S-Sz (1994)]\;}} \define\sc{\hbox{[S-Ch (1987)]\;}} \define\bus{\hbox{[B-S (1973)]\;}} \define\blps{\hbox{[BLPS (1992)]\;}} \define\dist{\text{dist\;}} \define\nt{\text{int\;}} \define\vtm{v^{t^-}} \define\vTm{v^{T^-}} \define\vtp{v^{t^+}} \define\vTp{v^{T^+}} \define\wtm{w^{t^-}} \define\wtp{w^{t^+}} \define\wTm{w^{T^-}} \define\wTp{w^{T^+}} \def \t {\text {\bf T}} \def \r {\Bbb R} \def \q {\text {\bf Q}} \def \z {\text {\bf Z}} \def \T {\text {\bold {T}}} \def \tf {\bold {T}^4} \def \qf {(q_1,q_2,q_3,q_4)} \define\sinf{S^{(-\infty,\infty)}x_0\,} \define\xim{$x_0\in M_d$\,} \define\mpp{M_{P_1,P_2}\,} \define\fcf{F_{-}\cap F_{+}\,} \define\sncx{$S^{(-\infty ,0)}_{C_i(P_1)}x_0$\,} \define\spcx{$S^{(0,\infty )}_{C_i(P_2)}x_0$\,} \define\sncy{$S^{(-\infty ,0)}_{C_i(P_1)} y$\,} \define\spcy{$S^{(0,\infty )}_{C_i(P_2)} y$\,} \define\steb{\left\{ S^t_{1,2}\right\}\,} \define\sthb{\left\{ S^t_{3,4}\right\}\,} \define\stnb{\left\{ S^t_{2,3,4}\right\}\,} \define\ste{S^t_{1,2}\,} \define\sth{S^t_{3,4}\,} \define\stn{S^t_{2,3,4}\,} \define\pikv{\pi^V_{2,3,4}\,} \define\sni{S^{(-\infty,0)}\,} \define\snis{S^{(-\infty,0)}_*\,} \define\snin{S^{(-\infty,0)}_{2,3,4}\,} \define\spi{S^{(0,\infty)}\,} \define\spis{S^{(0,\infty)}_*\,} \define\spie{S^{(0,\infty)}_{1,2}\,} \define\spih{S^{(0,\infty)}_{3,4}\,} \define\spin{S^{(0,\infty)}_{1,3,4}\,} \define\vk{(v_1,v_2,v_3,v_4)\,} \define\qvi{(q_1,v_1,I-v_1)\,} \define\qvk{(q_1,q_2,v_1,v_2)\,} \define\qvio{(q_1^0,v_1,I-v_1)\,} \define\qviz{(q_1^0, v_1^0, I-v_1^0)\,} \define\bbb{\bold {B}} \define\aaa{\bold {A}} \define\Q{\bold {Q}} \define\qm{\partial \bold {Q}^-} \define\qp{\partial \bold {Q}^+} \define\qi{\partial \bold {Q}_i} \define\mi{\partial {M}_i} \define\mpl{\partial {M}^+} \define\mm{\partial {M}^-} \define\suf{sufficient\ \ } \define\erg{ergodicity\ \ } \define\traj{S^{[a,b]}x_0} \define\flow{\left(\bold{M},\{S^t\},\mu\right)} \define\proj{\Bbb P^{\nu-1}(\Bbb R)} \define\ter{S^{d-1}\times\left[\Bbb P^{\nu-1}(\Bbb R)\right]^m} \define\sphere{S^{d-1}} \define\pont{(V;h_1,h_2,\dots ,h_m)} \define\projm{\left[\Bbb P^{\nu-1}(\Bbb R)\right]^m} \define\pontg{(V_0;g_1,g_2,\dots ,g_m)} \define\szorzat{\prod\Sb i=1\endSb\Sp m\endSp \Cal P_i} \define\szorzatk{\prod\Sb l=1\endSb\Sp k\endSp \Cal P_{i_l}} \define\ball{\overline{B}_{\varepsilon_0}(x_0)} \define\qv{(Q,V^+)} \document {\catcode\@=11\gdef\logo@{}} \advance\baselineskip 8pt \noindent July 26, 1995 \bigskip \bigskip \centerline{\bf The K-Property of Hamiltonian Systems with Restricted Hard Ball Interactions} \bigskip \centerline{N\'andor Sim\'anyi$^1$ and Domokos Sz\'asz\footnote{Research supported by the Hungarian National Foundation for Scientific Research, grant No. 1902}} \hbox{} \centerline{Mathematical Institute of the Hungarian Academy of Sciences} \centerline{H-1364, Budapest, P. O. B. 127, Hungary} \bigskip \bigskip \heading 1. Introduction \endheading \bigskip In 1985, Chernov and Sinai suggested the study of a mechanical model of elastic hard balls: the so-called {\it pencase}, which we will call the Chernov-Sinai pencase (cf. the manuscript [S-Ch(1985)]). Here large hard ball particles move on a torus, elongated in one spatial dimension; since then the cyclic order of the particles remains invariant, each ball can only interact with its two neighbours. Their idea was that, on one hand, because of the restricted form of the interactions, the expected ergodicity of the model could be easier to establish than that of general hard ball systems on tori and, on the other hand, this quasi-one-dimensional model seems to be a quite suitable mechanical one for the study of the hydrodynamic limit transition (cf. [Sin(1985)]). For systems of elastic hard balls on a torus Yakov Sinai, in 1963, [Sin(1963)] gave a mathematically rigorous version of Boltzmann's celebrated ergodic hypothesis: {\it the system of an arbitrarily fixed number $N$ of elastic hard balls moving in the $\nu-$torus $\Bbb T^\nu\ \ (\nu \ge 2)$ is ergodic --- of course, on the submanifold of the phase space specified by the trivial conservation laws}. The aforementioned hypothesis, which we call the {\it Boltzmann-Sinai Ergodic Hypothesis}, was first established for the case $N=\nu=2$ in Sinai's celebrated paper [Sin(1970)]. After several steps ( [S-Ch(1987)]: the case $N=2, \nu \ge 2$, [K-S-Sz(1991)]: the case $N=3, \nu \ge 2$, [K-S-Sz(1992)]: the case $N=4, \nu \ge 3$) the most general result so far was obtained by Sim\'anyi [Sim(1992)], who could verify the conjecture for the case $\nu \ge N \ge 2$ (a more detailed survey of the history of Boltzmann's ergodic hypothesis can be found in [Sz(1995)]). The aim of the present work is to verify the K-property of models closely related to the mechanical system of hard balls. The restriction we make is on the {\it graph of interactions}: we first assume that it is a tree whose {\it edge-degree} is bounded from above by a constant $D$. Let us make here two remarks: \roster \item "(i)" the {\it graph of interactions} means that only pairs of particles connected by an edge in this graph can interact (in our model they do this through a hard ball collision) whereas non-connected pairs do not interact at all; \item "(ii)" the {\it degree of an edge} of a graph is the number of edges adjacent to the given edge, and then the {\it edge-degree} of a graph is the maximum of the degrees of all edges. \endroster (The precise description of our model will be given soon). Then we prove that such a system is a K-mixing one if $\nu \ge D +2$, and, moreover, for the case $\nu=D+1$ we obtain that none of the relevant Lyapunov-exponents vanishes and the ergodic components are open. By strengthening the methods used in treating the aforementioned tree-interactions, (in particular, the geometric-algebraic part (1) of our strategy detailed at the end of this section), we can also treat an interaction graph which is not a tree: a {\it circle of length $N$}. This result implies the K-property of the Cher\-nov-Sinai pencase in dimension $\nu =4$. It is easy to see that in the Chernov-Sinai pencase, in dimensions $\nu \le 4$, the graph of interactions is a simple cyclic one (i. e. the graph that consists of just one circle of length $N$). Our methods also imply that the Chernov-Sinai pencase has non-zero Lypaunov-exponents and open ergodic components in our physical space dimension $\nu = 3$ (the pencase as a system of ball-particles loses order invariance if $\nu \ge 5$, and if one wants to maintain this property, systems of particles with other, less natural shapes can be, in fact, introduced and treated by our methods). It is worth mentioning that in the low-dimensional cases (starting from $\nu = 4$) our results provide the first mechanical models where the ergodicity of the system is established for an arbitrary number of degrees of freedom --- apart from a model introduced in [B-L-P-S(1992)], where the boundary of the vessel also contributes to the hyperbolic effects. Furthermore, an important advantage of the hereby presented method for going down to physical dimensions is that it avoids the use of more involved algebraic tools that seem unavoidable in the case of treating hard sphere systems in the general framework. Let us be more specific: introduce the model and formulate our results. Assume that, in general, a system of $N$ ($\ge 2$) balls, identified as $1, 2, \dots, N$ are given in $\bt^\nu$, the $\nu$-dimensional unit torus $(\nu\geq 2)$. Denote the phase point of the $i$th ball by $(q_i,v_i)\in \bt^\nu\times \Bbb R^{\nu}$. Assume also that the graph $\Cal G:= (\Cal V, \Cal E)$ of interactions is given, where the set of edges $\Cal E$ makes a {\it non-oriented tree} (i. e. a connected graph without loop) on the set of vertices $\Cal V:= \{1, 2, \dots, N\}$. (In the case of the mechanical model of $N$ hard balls on $\bt^\nu$ this graph is, of course, the complete one.) Further, for every $\{i,j\}:\ 1\le i,\ j\le N$, $i\ne j$, the potential functions $U_{i,j}:\ \Bbb R_+ \rightarrow \Bbb R_+ \cup \{\infty\}$ are given as follows $$U_{i,j}(r) = \cases \infty \qquad 0\le r < d_{i,j} \\ \ 0 \qquad\ \ r \ge d_{i,j} \endcases$$ where the numbers $d_{i,j}=d_{j,i}$ are nonnegative. They determine the set of edges $\Cal E$: $\{i,j\}\in\Cal E$ if and only if $d_{i,j}>0$. Moreover, the interaction between the pair $\{i,j\}$ of particles with configuration coordinates $q_i,q_j \in \Bbb T^\nu$ is $$\tilde U_{i,j}(q_i,q_j) = U_{i,j} (\Vert q_i-q_j\Vert ). \tag 1.1$$ where $\Vert . \Vert$ denotes the euclidean distance in the torus $\Bbb T^\nu.$ The configuration space $\overline{\bold Q}$ of the $N$ balls is a subset of $\bt^{N\cdot\nu}$: for every $\{i,j\} \in \Cal E$ we cut out from $\bt^{N\cdot\nu}$ the cylindric scatterer: $$\overline{C}_{i,j} =\left\{ Q=(q_1,\dots,q_N)\in \bt^{N\cdot\nu}: \left\Vert q_i-q_j\right\Vert < d_{i,j} \right\}. \tag 1.2$$ In other words $\overline{\boq}:= \bt^{N\nu}\setminus \bigcup_{\{i,j\} \in \Cal E} \overline{C}_{i,j}$ (one cylindric scatterer $\overline{C}_{i,j}$ can obviously consist of several parallel cylinders, due to the non-uniqueness of the center of mass in a torus). The energy $H={1\over 2}\, \sum^N_1 ||v_i||^2$ and the total momentum $P=\sum^N_1 v_i$ are first integrals of the motion. Thus, without loss of generality, we can assume that $H={1\over 2}$, $P=0$, and, moreover, that the sum of spatial components $B=\sum^N_1 q_i$ is zero. (If $P\not= 0$, then the center of mass has an additional conditionally periodic or periodic motion.) For these values of $H,P$ and $B$, the phase space of the system reduces to the unit tangent bundle $\bold{M}:=\bold{Q}\times S^{N\cdot\nu -\nu -1}$ of the configuration space $\bold Q$, where $$\bold {Q} := \left\{ Q\in \overline{\bold{Q}} :\sum^N_1 q_i =0\right\}$$ with $d:=\text{dim}\bold{Q} =N\cdot\nu -\nu$, and further $S^k$ denotes, in general, the $k$-dimensional unit sphere. It is easy to see that the dynamics of the $N$ particles, determined by their uniform motion with elastic collisions (corresponding to the interaction potentials (1.1)) between pairs belonging to $\Cal E$ on one hand, and the billiard flow $\{S^t:\, t\in \Bbb R\}$ in $\bold {Q}$ with specular reflections at $\partial \bold {Q}$ on the other hand, conserve the Liouville measure $d\mu = const\cdot dq\cdot dv$, and are isomorphic. These -- isomorphic -- dynamical systems $\left(\bold{M}, S^\Bbb R := \{S^t:\, t\in \Bbb R\}, \mu\right)$ are called {\it standard billiard ball systems with $\Cal D$-interactions} where $\Cal D:=(d_{i,j})_{i,j=1}^N$. It is evident that $\Cal D$ determines $\Cal E$ and thus the model, too. (These standard billiard ball flows of $N$ particles with $\Cal D$-interactions in $\Bbb T^\nu$ are, in fact, {\it semi-dispersing billiards} $(\bold M, S^\Bbb R, \mu)$, i. e. billiards with convex scatterers.) We also recall that a billiard is called {\it dispersing} if its scatterers are strictly convex (for instance, the standard billiard ball flow in the case of $N=2$ balls on $\Bbb T^\nu$). The first main result of our paper is the following \proclaim{Theorem A} Assume that the edge-degree of the tree of interactions is $D$, and for some $N\ge 2$, $\nu \ge D + 2$\ \ $\text{Int} \bold Q$ (or, equivalently $\text{Int} \bold M$) is connected. Then the standard billiard ball system $(\bold{M},\{ S^t\},\mu)$ with a $\Cal D$-interaction is a K-flow. \endproclaim \subheading{\bf Note 1} In general, if $\text{Int}\bold M$ decomposes into a finite number of connected components, then our methods provide that the standard billiard ball flow with a $\Cal D$-interaction is a K-system on each of these connected components. \subheading{\bf Note 2} As it has been proved in recent manuscripts by N. I. Chernov and C. Haskell, [Ch-H(1995)] on one hand, and by D. Ornstein and B. Weiss, [O-W(1995)] on the other hand, the K-mixing property of a semi-dispersing billiard flow actually implies its Bernoulli property, as well. Our second main result concerns the so-called {\it cyclic interaction}, where \break $\Cal E_c:= \{\{1,2\}, \{2,3\}, \dots, \{N-1,N\}, \{N,1\}\}$. Let $\Cal D_c$ be a corresponding matrix of interaction ranges. \proclaim {Theorem B} The standard billiard ball system with a cyclic interaction is a K-system if $\nu\ge 4$ and $\text{Int} \bold Q$ is connected. \endproclaim As mentioned before, an interesting mechanical realization of the cyclic interaction is the Chernov-Sinai pencase. Here the torus is an elongated one, i. e. (physical) balls of radii $r$ move on a torus $\bt^\nu_L:= \Bbb R^\nu / (L\Bbb Z \times \Bbb Z^{\nu-1})$ (in other words, $d_{i,j} = 2r$) for every $\{i,j\} \in \Cal E_c$. Now, as a particular case of Theorem B, we obtain \proclaim {Corollary} The Chernov-Sinai pencase is a K-system if $\nu = 4$ and $\sqrt {3}/4 < r < 1/2$. \endproclaim We note that actually the phase space of the pencase consists of exactly $N!$ connected, isomorphic components whenever $\nu \le 4$ and $\sqrt {\nu-1}/4 < r < 1/2$. Thus, of course, the K-property can only hold, and is meant here, on each of them, separately. By fortifying the inductive arguments we also obtain the following results: \proclaim{Theorem C} Assume that the edge-degree of the graph of interaction is $D$. If $N\ge 2$, $\nu = D + 1$, then, almost everywhere, none of the relevant Lyapunov-exponents of the standard billiard ball system $(\bold{M},\{ S^t\},\mu)$ with a $\Cal D$-interaction vanishes. Furthermore, the ergodic components of the system are open (and thus of positive measure) and on each of them the flow has the K-property. \endproclaim \proclaim {Theorem D} For the models of Theorem B and its Corollary, in dimension $\nu = 3$, almost everywhere, none of the relevant Lyapunov exponents vanishes. Furthermore, the ergodic components of the system are open (and thus of positive measure) and on each of them the flow has the K-property. \endproclaim \medskip {\bf General strategy.} The basic notion in the theory of semi-dispersing billiards is that of the {\it sufficiency} of a phase point or, equivalently, of its orbit. The conceptual importance of sufficiency can be explained as follows (for a technical introduction and our prerequisites, see section 2): In a suitably small neighbourhood of a (typical) phase point of a dispersing billiard the system is {\it hyperbolic}, i. e. its relevant Lyapunov exponents are not zero. For a semi-dispersing billiard the same property holds for sufficient points only! Physically speaking, a phase point is sufficient if its trajectory encounters in its history all possible degrees of freedom of the system. Then the fundamental theorem of semi-dispersing billiards (see the main result of [S-Ch (1987)]) says that --- under certain conditions --- a suitably small neighbourhood of a sufficient phase point does belong to one ergodic component. As a consequence, if, for instance, almost every phase point of a \sdb\ is sufficient, then this property implies that the ergodic components are open and, therefore, their number is countable. To obtain, however, (global) \erg of the flow, a more stringent property of the subset of non-\suf points is needed. For that purpose, in our work with A. Kr\'amli \kssa, the topological property of {\it slimness} (earlier misleadingly called 'residuality'), closely related to that of {\it topological codimension two}, was suggested and used in later works, too. \proclaim {Definition 1.3} We say that a subset of a smooth manifold $M$ is {\rm slim} if it can be covered by a countable union of codimension two (i. e. at least two), closed subsets of $\mu-$measure zero, where $\mu$ is a smooth measure on $M$. \endproclaim Here the dimension of a separable metric space is its topological or Hurewicz dimension, which is any one of the following equivalent notions: the small inductive, the large inductive, or the covering dimension, see, for instance, [H-W(1941)]. It has long been an accepted idea among experts that the right way for proving the K--property of hard ball systems is an induction on the number $N$ of balls. According to our strategy initiated in \kssc and explained in the introductions of \kssd and \sima, there are three major parts in such an induction, once a combinatorial property, called {\it richness} of the symbolic collision sequence of a trajectory, had been suitably defined. (Our general concept of richness, appropriate for hard ball systems, is introduced in Definition 3.2.) \roster \item"{(1)}" The geometric--algebraic considerations'' on the codimension of the closed algebraic sets describing the non--sufficient trajectory segments with a combinatorially rich symbolic collision structure (This step has no inductive character!); \item"{(2)}" Proof of slimness (negligibility) of the set of phase points with a combinatorially non--rich collision sequence by using the K--property of less than $N$ hard balls in the torus; \item"{(3)}" Checking the Chernov--Sinai Ansatz --- an important necessary condition for the proof of the Theorem on Local Ergodicity, see Condition (A) before Theorem 5 in [S-Ch (1987)], or Condition 3.1 in \kssb. This step also uses the K--property of less than $N$ balls and step (1) for the system of $N$ hard balls in the torus. \endroster \bigskip The paper is organized as follows: Section 2 summarizes certain prerequisites. Section 3 provides the proofs of the theorems and their corollaries assuming Main Lemmas 3.4 and 3.7 of geometric-algebraic nature. Then Main Lemma 3.4 related to tree-interactions and used in the proofs of Theorems A and C is established in Section 4, while Main Lemma 3.7 related to the cyclic interaction and used in the proofs of Theorems B and D gets verified in Section 5. Section 6 contains some further remarks. \bigskip \bigskip \heading 2. Prerequisites. \endheading \bigskip As to the basic notions concerning semi-dispersing billiards we refer to the paper [K-S-Sz (1990)]. For convenience and brevity, we will throughout use the concepts and notations of sections 2 and 3 of the aforementioned work. Here we only summarize some further notions from \kssc, \kssd, [Sim(1992)]. These are either new or their exposition is simpler than that given in the original work. An often used abbreviation is the shorthand $S^{[a,b]}x$ for the trajectory segment $\{S^tx: \, a\le t\le b\}$. The natural projections from $\bold{M}$ onto its factor spaces are denoted, as usual, by $\pi: \bold{M}\rightarrow \boq$ and $p:\ \bold{M}\rightarrow S^{N\cdot\nu-\nu-1}$ or, sometimes, we simply write $\pi(x)=Q(x)=Q$ and $p(x)=V(x)=V$ for $x=(Q,V)\in\bold{M}$. Any $t \in [a,b]$ with $S^tx \in \partial M$ is called {\it a collision moment or collision time}. As pointed out in previous works on billiards, the dynamics can only be defined for trajectories where the moments of collisions do not accumulate in any finite time interval (cf. Condition 2.1 of [K-S-Sz(1990)]). An important consequence of Theorem 5.3 of [V(1979)] is that --- for semi-dispersing billiards --- there are {\it no trajectories at all with a finite accumulation point of collision moments}. As a result, for an arbitrary non-singular orbit segment $S^{[a,b]}x$ of the standard billiard ball flow, there is a uniquely defined maximal sequence $a \le t_1 < t_2< \dots < t_m \le b:\ m \ge 0$ of collision times and a uniquely defined sequence $\sigma_1 <\sigma_2 < \dots < \sigma_m$ of colliding pairs'', i. e. $\sigma _k = \{ i_k, j_k \}$ whenever $Q(t_k)=\pi(S^{t_k}x) \in \partial \overline{C}_{i_k,j_k}.$ The sequence $\Sigma:= \Sigma(S^{[a,b]}x):= (\sigma_1, \sigma_2, \dots, \sigma_m)$ is called the {\it symbolic collision sequence} of the trajectory segment $S^{[a,b]}x$. As well known, billiards are dynamical systems with {\it singularities}. A collision at a point $x\in\partial M$ such that, in $\pi(x)$, at least two smooth pieces of $\partial{\text {\q}}$ meet is called a {\it multiple} collision. A collision is called {\it tangential} if $x\in\partial M$ and $p(x)\in \Cal {T}_{\pi(x)}\partial{\bold {Q}}$, i. e. $p(x)$ is tangential to $\partial \bold Q$ at the point of reflection. We shall use the collection $\sr$ of all singular reflections: \proclaim{Definition 2.1} The set $\sr$ is the collection of all phase points $x\in \partial\bold M$ for which the reflection, occurring at $x$, is {\rm singular} (tangential or multiple) and, in the case of a multiple collision, $x$ is supplied with the {\it outgoing} velocity $V^+$. \endproclaim It is not hard to see that $\sr$ is a compact cell--complex in $\bold M$ and $\dim (\sr )=\dim \bold M-2= 2d-3$. As usual, we will denote by $\bold M^*$ the set of phase points $x \in \bold M$ whose full trajectory contains infinitely many collisions such that at most one of them is singular. The subset of points $x \in \bold M^*$ whose orbit has no singular collision at all will be denoted by $\bold M^0$, and finally we denote $\bold M^1:= \bold M \setminus \bold M^0$. \bigskip \heading Neutral Subspaces, Advance and Sufficiency \endheading \medskip Consider a {\it non--singular} trajectory segment $S^{[a,b]}x$. Suppose that $a$ and $b$ are {\it not moments of collision}. Before defining the neutral linear space of this trajectory segment, we note that the tangent space of the configuration space $\boq$ at interior points can be identified with the common linear space $$\Cal Z=\left\{(w_1,w_2,\dots ,w_N)\in (\Bbb R^{\nu})^N: \, \sum \Sb i=1 \endSb \Sp N \endSp w_i=0 \right\}. \tag 2.1$$ \proclaim {Definition 2.2} The neutral space $\Cal N_0(S^{[a,b]}x)$ of the trajectory segment $S^{[a,b]}x$ at time zero (a<00) \; \text{s. t.} \; \forall \alpha \in (-\delta,\delta) \\ & p\left(S^a\left(Q(x)+\alpha W,V(x)\right)\right)=p(S^ax) \, \& \\ & p\left(S^b\left(Q(x)+\alpha W,V(x)\right)\right)=p(S^bx) \big \}. \endaligned $$\endproclaim It is known (see (3) in Section 3 of [S-Ch (1987)]) that \Cal N_0(S^{[a,b]}x) is a linear subspace of \Cal Z indeed, and V(x)\in \Cal N_0(S^{[a,b]}x). The neutral space \Cal N_t(S^{[a,b]}x) of the segment S^{[a,b]}x at time t\in [a,b] is defined as follows:$$ \Cal N_t(S^{[a,b]}x)=\Cal N_0\left(S^{[a-t,b-t]}(S^tx)\right). \tag 2.3 $$It is clear that the neutral space \Cal N_t(S^{[a,b]}x) is canonically identified with \Cal N_0(S^{[a,b]}x) by the usual identification of the tangent spaces of \boq along the trajectory S^{(-\infty,\infty)}x (see, for instance, Section 2 of \kssb). (Intuitively speaking, the neutral subspace is the orthocomplement of the positive subspace of the second fundamental form of the image of a parallel beam of light). Our next definition is that of the {\it advance}. Consider a non-singular orbit segment S^{[a,b]}x with a symbolic collision sequence \Sigma=(\sigma_1, \dots, \sigma_m)\ (m \ge 1) as at the beginning of the present section. For x=(Q,V) \in \bold M and W\in\Cal Z, \Vert W\Vert sufficiently small, denote T_W(Q,V):=(Q+W,V). \proclaim {Definition 2.4} For any 1 \le k \le m and t \in [a,b], the advance A(\sigma_k):\ \Cal N_t(S^{[a,b]}x) \rightarrow \Bbb R is the unique linear extension of the linear functional defined in a sufficiently small neighbourhood of the origin of \Cal N_t(S^{[a,b]}x) in the following way:$$ A(\sigma_k)(W):= t_k(x)-t_k(S^{-t}T_WS^tx). $$\endproclaim It is now time to bring up the basic notion of {\it sufficiency} of a trajectory (segment). This is the utmost important necessary condition for the proof of the Theorem on Local Ergodicity for Semi--Dispersing billiards, see Condition (ii) of Theorem 3.6 and Definition 2.12 in \kssb. \proclaim{Definition 2.5} \roster \item The non--singular trajectory segment S^{[a,b]}x (a and b are supposed not to be moments of collision) is said to be {\it sufficient} if and only if the dimension of \Cal N_t(S^{[a,b]}x) (t\in [a,b]) is minimal, i.e. \text{dim}\ \Cal N_t(S^{[a,b]}x)=1. \item The trajectory segment S^{[a,b]}x containing exactly one singularity is said to be {\it sufficient} if and only if both branches of this trajectory segment are sufficient. \endroster \endproclaim For the notion of trajectory branches see, for example, the end of Section 2 in [Sim(1992)-I]. \proclaim{Definition 2.6} The phase point x\in \bold M with {\it at most one singularity} is said to be sufficient if and only if its whole trajectory S^{(-\infty,\infty)}x is sufficient, which means, by definition, that some of its bounded segments S^{[a,b]}x is sufficient. \endproclaim In the case of an orbit S^{(-\infty,\infty)}x with exactly one singularity, sufficiency requires that both branches of S^{(-\infty,\infty)}x be sufficient. \bigskip \heading Weak Lemma on Avoiding of Balls \endheading The next lemma is a measure-theoretic version of Lemma 3 in [K-S-Sz(1989)], and is slightly more generally formulated here than Lemma 2.16 of [K-S-Sz(1991)]. We consider a mixing semi-dispersing billiard in a d-dimensional compact domain {\q}, where {\q}\subset {\r}^d or {\q}\subset {\t}^d. Let \emptyset\ne B\subset {\q} be an open subset of \bold M and let H\subset {\r} be a subset of {\r} such that sup H= +\infty. We introduce the notation$$ A_H(B):=\{x \in \bold M: S^tx \not\in B\; \text {\rm for \, \, every}\; t\in H\}. $$\noindent Here A_H(B) is the set of those phase points of \bold M whose trajectory avoids the ball B at every time t\in H. \vskip0.5cm \bf Lemma 2.7. \it (Weak lemma on avoiding of balls):$$\mu (A_H(B))=0,$$\noindent \rm where \mu is the usual invariant measure of the billiard system (M,\{ S^t\},\mu ).\vskip0.5cm \heading Decompositions and sub-billiards \endheading It is intuitively clear that trajectory segments along which the system of balls decomposes into at least two non-interacting subclasses can not gather all degrees of freedom or, in a more technical wording, the segment cannot be sufficient. Throughout the paper we will be using the following notations: Assume that a system \kn = \{1, 2, \dots, N\} of N\ (\ge 2) balls is decomposed into two non-empty classes: \kn = P_1 \cup P_2, \ P_1\cap P_2 = \emptyset, \vert P_i \vert \neq 0 \ (i = 1,2). We say that the trajectory segment S^{(a,b)}x {\it is partitioned} by P = \{P_1,P_2\} if all of its non-tangential collisions occur among particles of the same class of P, only. In such a case, the action of S^t on x = \{\tilde x_{P_1},\tilde x_{P_2}\} (t\in [a,b]) is certainly the product of two independent {\it subdynamics}: \tilde S^t_{P_1} {\tilde x}_{P_1} and \tilde S^t_{P_2} {\tilde x}_{P_2}. To be more exact, we express S^tx in detail as a direct product. To this end decompose$$ x = \{\{(q_j,v_j):\ j \in P_1 \},\{(q_j,v_j):\ j \in P_2 \}\} := \{\tilde x_{P_1},\tilde x_{P_2}\}. $$Denote$$ \sum _{j \in P_i} q_j = \bar q_i,\quad \sum _{j \in P_i} v_j = I_i,\quad \frac{1}{2} \sum _{j \in P_i} v_j^2 = E_i $$for i=1,2. By a standard change of coordinates \coK, to every \tilde x_{P_i} there corresponds an element x_{P_i} = \coK \tilde x_{P_i} \in M_{P_i}, (the phase space of a billiard with \vert P_i \vert balls), and S^tx can be understood as$$ S^tx = \{ \coKt^{-1}S^t_{P_i}\coK \tilde x_{P_i} : \ i =1,2\}. $$The dynamics S^t_{_i} or \coKt^{-1}S^t_{P_i}\coK acting on \tilde x_{P_i} will be called {\it subdynamics} or a {\it sub-billiard}. \bigskip \heading 3. Proof of the Theorems Assuming Main Lemmas 3.4 and 3.7 \endheading \bigskip \proclaim {Definition 3.1} We say that the symbolic collision sequence \Sigma=(\sigma_1, \dots, \sigma_m) is {\sl connected} if the collision graph of this sequence: \Cal G_\Sigma:= (\Cal V= \{1,2, \dots, N\}, \Cal E_\Sigma:= \{\{i_k,j_k\}:\ \text{where}\ \sigma_k = \{i_k, j_k\},\ 1 \le k \le m\}) is connected. \endproclaim \proclaim {Definition 3.2} We say that the symbolic collision sequence \Sigma=(\sigma_1, \dots, \sigma_m) is {\sl C-rich}, with C being a natural number, if it can be decomposed to at least C consecutive, disjoint collision sequences in such a way that each of them is connected. \endproclaim {\bf Proof of Theorem A.} Let us fix a system satisfying the conditions of the Theorem where the number of particles is N\ge 2. The proof of the K-property of the standard billiard flow with \Cal D-interaction will go on by induction on the number N of particles. Assume thus that the standard billiard ball system with every permitted \Cal D-interaction and in every permitted dimension has the K-property for any number of particles n1, and suppose the inductive hypothesis for smaller values of r-p. Plainly, we can assume that \sigma_k\ne\sigma_p for k=p+1,\dots,r-1. Write down the Connecting Path Formula (CPF, Lemma 2.9 of [Sim(1992)-II]) for the new advance A_r in terms of A_p and the advances A_k with pl, that is, there is a natural time ordering among the classes C_i containing the edges adjacent to i_p (j_p). \endproclaim \subheading{Proof} A straightforward consequence of the definition of the equivalence relation \sim. \qed We can now finish the proof of Lemma 4.2. Suppose that, contrary to the statement of the lemma, A_q(n_0)\ne 0, A_p(n_0)=A_r(n_0)=0 with some vector n_0\in\Cal N_a\left(\traj\right). Evaluate (4.8) at this vector n_0:$$ \sum\Sb i=1\endSb\Sp s\endSp A(C_i)[n_0]\cdot \Gamma(C_i)=0. \tag 4.10 $$Since A(C_i)[n_0]\ne 0 for \sigma_q\in C_i, the equation (4.10) means that the vectors$$ \Gamma(C_1),\dots,\Gamma(C_s) \; (\in\Bbb R^{\nu}) $$are linearly dependent. According to the previous sub-lemma and the fact that the graph \Cal G is a tree, the velocity changes \Gamma(C_1),\dots,\Gamma(C_s) may be chosen independently. Taking into account the inequality s\le D, we have that the linear dependence (4.10) can only occur on a set of phase points x_0 with codimension at least \nu+1-D. Hence Lemma 4.2. \qed \medskip \heading Finishing the proof of Main Lemma 3.4 \endheading \medskip Suppose that the subgraphs \Sigma^{(1)}=(\sigma_1,\dots,\sigma_k), \Sigma^{(2)}=(\sigma_{k+1},\dots,\sigma_l), and \Sigma^{(3)}=(\sigma_{l+1},\dots,\sigma_m) of \Sigma are all connected on the vertex set \Cal V=\{1,2,\dots,N\}, i. e. the set of edges of \Sigma^{(j)} (j=1,2,3) is the set \Cal E of all edges of \Cal G. The fact that \Cal G is a tree and Lemma 4.2 straightforwardly imply that all advances A_1,\dots,A_m must be equal modulo a closed algebraic set of exceptional phase points x_0 with codimension at least \nu-D. The identity of these advances, however, precisely means that the orbit segment \traj is sufficient, see also Lemma 2.13 of [Sim(1992)-II]. This finishes the proof of Main Lemma 3.4. \qed \bigskip \bigskip \heading 5. Trajectories with a Circular Collision Graph. \\ The Proof of Main Lemma 3.7 \endheading \bigskip In this section we slightly generalize the results of the previous paragraph by considering the following graph \Cal G_c=(\Cal V,\Cal E_c) of the {\it circular} interaction:$$ \Cal E_c=\left\{\{i,i+1\}|\, 1\le i\le N-1\right\}\cup\left\{\{1,N\}\right\}. \tag 5.1 $$For this graph \Cal G_c, which is a circle of length N, the edge degree D is equal to two. The actual importance of this circular interaction is explained by the fact that this is just the interaction of the particles in the so called {\it pencase model} introduced by Chernov and Sinai, see [S-Ch(1985)] or Section 1. The content of this section is exclusively the proof of Main Lemma 3.7. Before proving anything, however, it is worth noting that the whole issue of proving sufficiency by the use of combinatorial richness is totally insensitive of the uniform velocity dilations of the billiard flow. Therefore, it is not necessary to assume the usual energy normalization \sum_{i=1}^N v_i^2=1, so we will not do that. First of all, we need the following \proclaim{Lemma 5.2} Suppose that the graph of allowed collisions is the circle \Cal G_c and \traj is a non-singular orbit segment with the symbolic sequence \Sigma=(\sigma_1,\dots,\sigma_m). If 1\le p1 and the statement is true for smaller values of r-p. Just like in the previous section, we can assume that \sigma_k\ne\sigma_p for k=p+1,\dots,r-1, and \sigma_p=\{1,N\}, i. e. i_p=1, j_p=N. The useful, short form of the Connecting Path Formula for the new advance A_r in terms of the earlier advances is now$$ \aligned A_r\cdot\left(v_1^-(t_r)-v_N^-(t_r)\right)&= A_p\cdot\left(v_1^+(t_p)-v_N^+(t_p)\right) \\ +A_{\{1,2\}}\cdot\sum\Sb k\\ p0 whereas the remaining $N-1$ ones are point particles, the tree of interactions is star-like, i. e. $\Cal E=\{\{1,i\}: 2 \le i \le N\}$. 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