Open billiards : invariant and conditionally invariant probabilities on cantor sets

Abstract

Billiards are the simplest models for understanding the statistical theory of the dynamics o f a gas in a closed compartment. We analyze the dynamics of a class o f billiards ( the open billiard on the plane) in terms of invariant and conditionally invariant probabilities. The dynamical system has a horseshoe structure. The stable and unstable manifolds are analytically described. The natural probability 1-' is invariant and has support in a Cantor set. This probability is the conditional limit of a conditional probability 1-'F that has a density with respect to the Lebesgue measure. A formula relating entropy, Lyapunov exponent, and Hausdorff dimension of a natural probability 1-' for the system is presented. The natural probability 1-' is a Gibbs state of a potential '1/J (cohomologous to the potential associated to the positive Lyapunov exponent; see formula (0.1)), and we show that for a dense set of such billiards the potential '1/J is not lattice. As the system has a horseshoe structure, one can compute the asymptotic growth rate of n(r), the number of closed trajectories with the largest eigenvalue of the derivative smaller than r. This theorem implies good properties for the poles of the associated Zeta function and this result turns out to be very important for the understanding of scattering quantum billiards.