Stable laws for chaotic billiards with cusps at flat points

TL;DR

This paper studies billiard systems with flat points at cusps and demonstrates that normalized sums of observables follow a skewed alpha-stable distribution, revealing new statistical properties of such chaotic systems.

Contribution

It establishes the convergence of Birkhoff sums to an alpha-stable law for billiards with flat cusps, a novel result in the statistical analysis of these systems.

Findings

Normalized Birkhoff sums converge to an alpha-stable law

The system exhibits heavy-tailed stable distribution behavior

Chaotic billiards with cusps have distinct statistical properties

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 H\"older 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.