01-246 P. B\'alint, N. Chernov, D. Sz\'asz, P. T\'oth

Multi-dimensional Semi-dispersing Billiards: Singularities and the Fundamental Theorem (747K, Postscript) Jul 6, 01

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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 we fix the problem. The fundamental theorem is now 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 -- hard ball systems, Lorentz gases with spherical scatterers, and Bunimovich-Reh\'a\v cek stadia.

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