Cylindric Billiards and Transitive Lie Group Actions

A conjecture is formulated and discussed which provides necessary and sufficient condition for the ergodicity of cylindric billiards (this conjecture improves a previous one of the second author). This condition requires that the action of a Lie-subgroup G of the orthogonal group O(d) (d being the dimension of the billiard in question) be transitive on the unit sphere Sd−1. If C1, . . . , Ck are the cylindric scatterers of the billiard, then G is generated by the embedded Lie-subgroups Gi of O(d), where Gi consists of all orthogonal transformations of R that leave the points of the generator subspace of Ci fixed (1 ≤ i ≤ k). In this paper we can prove the necessity of our conjecture and we also formulate some notions related to transitivity. For hard ball systems, we can also show that the transitivity holds in general: for arbitrary number N ≥ 2 of balls, arbitrary masses m1, . . . ,mN and in arbitrary dimension ν ≥ 2. This result implies that our conjecture is stronger than the Boltzmann-Sinai ergodic hypothesis for hard ball systems. As a by-remark, we can give a somewhat surprising characterization of the positive subspace of the second fundamental form for the evolution of special orthogonal manifold (wavefront), namely for the parallel beam of light. Thus we obtain a new characterization of sufficiency of an orbit segment. 1Research supported by the Hungarian National Foundation for Scientific Research, grants OTKA-7275 and OTKA-16425.