STATISTICAL PROPERTIES OF THE PERIODIC LORENTZ GAS - MULTIDIMENSIONALCASE

In 1981 Bunimovich and Sinai established the statistical properties of the planar periodic Lorentz gas with finite horizon. Our aim is to ex tend their theory to the multidimensional Lorentz gas. In that case th e Markov partitions of the Bunimovich-Sinai type, the main tool of the ir theory, are not available. We use a crude approximation to such par titions, which we call Markov sieves. Their construction in many dimen sions is essentially different from that in two dimensions; it require s more routine calculations and intricate arguments. We try to avoid t echnical details and outline the construction of the Markov sieves in mostly qualitative, heuristic terms, hoping to carry out our plan in f ull detail elsewhere. Modulo that construction, our proofs are conclus ive. In the end, we obtain a stretched-exponential bound for the decay of correlations, the central limit theorem, and Donsker's Invariance Principle for multidimensional periodic Lorentz gases with finite hori zon.