Domokos Szász (Technical University of Budapest)

Algebraic methods and the ergodic hypothesis for hard balls.

Abstract. The Ergodic Hypothesis for Hard Balls says that the systems of N elastic hard balls moving on the d-torus is ergodic modulo the trivial invariants of motion. A simple observation is that this system is isomorphic to a billiard with convex obstacles (which are even strictly convex if N=2). For N=2 the hypothesis got settled by the Moscow school (Sinai, 1970, d=2; Chernov-Sinai, 1987, d > 2). Afterwords the Budapest school reached various results for N >2 (most notably Krámli, Simányi and Szász for N=3 and 4 and Simányi for d > N-1) by introducing dynamical-topological and geometric-algebraic methods.

In 1999, Simányi and Szász gave a partial solution of the Boltzmann-Sinai Ergodic Hypothesis by establishing that typical hard ball systems are hyperbolic, i. e. all their Lyapunov exponents are non-zero a. e.. Several experts, however, were surprised that their methods relied quite heavily on the fact that in the isomorphic billiard system the boundaries of the obstacles are quadratic algebraic manifolds (indeed, they are cylinders with spherical bases). Earlier, in the theory of hyperbolic billiards only some smoothness, C² in general, of these boundaries was assumed and used.

Nevertheless, more recently it turned out that the algebraicity of the obstacle boundaries seems to be an essential assumption even in the fundamentals of the theory of billiards with convex obstacles. In fact, here a useful property of algebraic submanifolds is that they can be decomposed into a finite number of nice Lipschitz graphs of functions (these recent results are joint with P. Bálint, N. Chernov and P. Tóth).