No abstract

We prove that the billiard map in a Bunimovich stadium has polynomial decay of correlations of the order of n-1. Ergodic Bunimovich-type billiards (two arcs not bigger than half a circle) that do not have trajectories with infinitely many consecutive bounces on straight lines satisfy the same property. We also prove that a very particular class of these billiards have polynomial decay of correlations of the order of n-2.

In this paper we present an improved version of our work on the fundamental theorem of Sinai and Chemov. It can be used to study the ergodicity of dynamical systems with singularities if an increasing non-degenerate quadratic form is defined almost everywhere. We prove the ergodicity of *e billiard map in the cardioid. The methods used in this proof allow one to check which of the billiards with Pesin region of measure one are ergodic.

Dispersing billiards introduced by Sinai are uniformly hyperbolic and have strong statistical properties (exponential decay of correlations and various limit theorems). However, if the billiard table has cusps (corner points with zero interior angles), then its hyperbolicity is nonuniform and statistical properties deteriorate. Until now only heuristic and experiments results existed predicting the decay of correlations as O(1/n). We present a first rigorous analysis of correlations for dispersing billiards with cusps.

This is an attempt to study mathematically billiards with moving baundaries. We assume that the boundary remains closed, regular and strictly mnvex. deforming periodically in time. in the normal direction. We describe the associated billiard diffenmorphism and the corresponding invariant measure. We discuss the stability of %periodic orbits and investigate the boundedness of the velocity in some precise examples. Finally, we present the Hamiltonian formalism and the symplectic structure, considering that a moving billiard is a billiard with rigid boundary on an augmented configuafion space, with a singular metric.