Abstract:

The system of N elastically colliding hard balls with masses M_i radius r, moving uniformly in the flat torus is considered. Since Sinai's celebrated 1970 paper, where he proved hyperbolicity and even ergodicity in the case of two, 2-dimensional discs, some pleasant, `exact' properties of this particular interaction have always been exploited in attacking the problem. With N. Simanyi, we can prove that the flow is completely hyperbolic for almost every (N+1)-tuple (m_1...,m_N;r) of the outer geometric parameters. Our new, algebraic method relies upon the very algebraic form of the equations describing the dynamics (in analogy with geometric constructions by a compass and a ruler).