Multi-Dimensional Semi-Dispersing Billiards: Singularities and the Fundamental Theorem

Abstract

The fundamental theorem (also called the local ergodic theorem) was introduced by Sinai and Chernov in 1987, see [S-Ch(1987)] and an improved version in [K-S-Sz(1990)]. It provides sufficient conditions on a phase point under which some neighborhood of that point belongs to one ergodic component. This theorem has been instrumental in many studies of ergodic properties of hyperbolic dynamical systems with singularities, both in 2-D and in higher dimensions. The existing proofs of this theorem implicitly use the assumption on the boundedness of the curvature of singularity manifolds. However, we found recently ([B-Ch-Sz-T(2000)]) that, in general, this assumption fails in multidimensional billiards. Here the fundamental theorem is established under a weaker assumption on singularities, which we call Lipschitz decomposability. Then we show that whenever the scatterers of the billiard are defined by algebraic equations, the singularities are Lipschitz decomposable. Therefore, the fundamental theorem still applies to physically important models -among others to hard ball systems, Lorentz gases with spherical scatterers, and Bunimovich-Reháček stadia. 452In several papers that appeared, singularities were assumed -either explicitly or implicitly -to consist of smooth 1-codim submanifolds of the phase space. Often, Vol. 3, 2002 Multi-Dimensional Semi-Dispersing Billiards 457

Flow-Invariant Hypersurfaces in Semi-Dispersing Billiards

“…Geometrically, this means that ∂D is strictly convex (resp., just convex) as seen from inside D. For a recent detailed studies of the dynamics in semi-dispersing billiards we refer the reader to [BCST03]. Now let J ⊂ D be a small smooth compact hypersurface (a submanifold of codimension-one with boundary), which is locally flow-invariant, i.e.…”

“…In this seminal paper Sinai also developed a general theory of planar dispersing billiards. Then Sinai and Chernov [SC87] extended these results to systems of N = 2 balls in any dimension d > 2, as well as to other multidimensional dispersing billiards (there was a notable oversight in [SC87] that was corrected later in [BCST02]). …”

Local ergodicity for systems with growth properties including multi-dimensional dispersing billiards

“…regular coverings of local neighborhoods with parallelopipeds, the Chernov-Sinai Ansatz etc. We do not consider the issue of this "non-uniformly hyperbolic" fundamental theorem here, we refer to the literature instead, see [SCh], [KSSz], [Ch1], [LW], [BChSzT2], [B] and refereces therein.…”

“…In [BChSzT1] a pathological behaviour of singularities in multi-dimensional (semi-)dispersing billiards was found. This discovery calls for a reconsideration of earlier proofs of ergodicity in multi-dimensional semi-dispersing billiards.…”

“…This discovery calls for a reconsideration of earlier proofs of ergodicity in multi-dimensional semi-dispersing billiards. The papers [BChSzT2] and [B] handle the problem for certain special cases: [BChSzT2] deals with the case of algebraic scatterers, while [B] treats strictly dispersing billiards with highly smooth scatterers, and a strong condition on the complexity of the singularities. The present paper also contributes to this work: in Theorem 4.4 we prove ergodicity for a broader class of billiards, requiring only C 3 smoothness of the scatterers, but still requiring strict dispersing, and a weaker assumption about complexity of singularities.…”