SIAM Journal on Applied Mathematics
Volume 56, Number 2
pp. 651-680
1996 Society for Industrial and Applied Mathematics

"Open billiards: invariant and conditionally invariant probabilities on Cantor sets"

Artur Lopes and Roberto Markarian

[Abstract:] Billiards are the simplest models for understanding the statistical theory of the dynamics of a gas in a closed compartment. We analyzethe dynamics of a class of billiards (the open billiard on the plane)in terms of invariant and conditionally invariant probabilities. Thedynamical 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 theconditional limit of a conditional probability $\mu_F$ that has adensity 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 naturalprobability $\mu$ 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 ofsuch billiards the potential $\psi$ 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 eigenvalueof the derivative smaller than $r$. This theorem implies good properties for the poles of the associated Zeta function and thisresult turns out to be very important for the understanding ofscattering quantum billiards.

Key words. open billiards, Cantor sets

AMS Subject Classifications. 58F13, 58F11

 


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