Dispersing Billiards with Cusps: Slow Decay of Correlations

Abstract

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 experimental results existed predicting the decay of correlations as \({\mathcal{O}(1/n)}\) . We present a first rigorous analysis of correlations for dispersing billiards with cusps.