Open billiards : cantor sets, invariant and conditionally invariant probabilities
Fecha
1994Materia
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 co ...
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. ...
En
Cadernos de matemática e estatística. Série A, Trabalho de pesquisa. Porto Alegre. N. 39 (ago. 1994), f. 1-31
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