Stable laws for chaotic billiards with cusps at flat points

Abstract

We consider billiards with a single cusp where the walls meeting at the vertex of the cusp have zero one-sided curvature, thus forming a flat point at the vertex. For Hölder continuous observables, we show that properly normalized Birkoff sums, with respect to the billiard map, converge in law to a totally skewed $\alpha$-stable law.

42 pages. The paper has been revised to improve presentation, correct typos, and fix an error in what is now Proposition 4.2. To appear in Annales Henri Poincaré