Open Billiards: Invariant and Conditionally Iinvariant Probabilities on Cantor Sets

Billiards are the simplest models for understanding the statistical theory of the dynamics of a gas in a closed compartment. We analyze the dynamics of a class of 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 $\mu $ is invariant and has support in a Cantor set. This probability is the conditional limit of a conditional probability $\mu _F $ that has a density with respect to the Lebesgue measure. A formula relating entropy, Lyapunov exponent, and Hausdorff dimension of a natural probability $\mu $ for the system is presented. The natural probability $\mu $ is a Gibbs state of a potential$\psi $ (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 tb is not lattice. As the system has a horseshoe structure, ...

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