Open billiards : cantor sets, invariant and conditionally invariant probabilities

Abstract

A b stract - Billiards are the simplest models fur nudcrstanding statistical properties of thc dynamics of a gas in a closed compartment. vVc analyze the dynamics of a class of billiards (the open billiard on the plane) in terms of iuva.riant and conditionally inva.riant proba.bilities. The dynamical system has a horse-shoe structure. The stable and unstable manifolds are analytically described. The natural probability J.L is invariant and has support in a Cantor set. This probability is the conditiona.l limit of a conditiona.l probability J.LF that has a Holder continuous density with respcct to the Lebesgue measure. A formula relating entropy, Liapunov exponent and Hausdorff dimcnsion of a natural probability J.L for the system is presented. The natural probability fl is a Gibbs state of a potcntial 'lj; ( cohomologous to the potential associated to the positive Liapunov exponent, see formula (0.1 )), and we show that for a dense set of such billiards the potential 'lj; is not lattice. As the system has a horse-shoe structure one can compute the asymptotic growth rate of n(1·), the number of closed trajectories with the largest eigcuvalue of the derivative smaller tKru1 .r.