P. B\'alint, N. Chernov, D. Sz\'asz, P. T\'oth
Multi-dimensional Semi-dispersing Billiards: Singularities and the Fundamental Theorem
(747K, Postscript)
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.