Convergence to $\alpha$-stable L\'evy motion for chaotic billiards with several cusps at flat points

TL;DR

This paper proves that for billiards with multiple cusps, normalized sums of observables converge to an $oldsymbol{ ext{alpha}}$-stable Lévy motion in the Skorokhod $M_1$-topology, extending previous results from single cusp cases.

Contribution

It extends convergence results from single symmetric cusps to multiple, possibly asymmetric cusps, allowing for more general skewness parameters and stronger functional convergence.

Findings

Convergence to $oldsymbol{ ext{alpha}}$-stable Lévy motion in $M_1$-topology.

Limits depend on curvature of flat points and observable values.

Stronger than previous one-point marginal results, but $J_1$-topology convergence is not possible.

Abstract

We consider billiards with several possibly non-isometric and asymmetric cusps at flat points; the case of a single symmetric cusp was studied previously in Zhang (2017) and Jung & Zhang (2018). In particular, we show that properly normalized Birkhoff sums of H\"older observables, with respect to the billiard map, converge in Skorokhod's $M_1$-topology to an $\alpha$-stable L\'evy motion, where $\alpha$ depends on the `curvature' of the flattest points and the skewness parameter $\xi$ depends on the values of the observable at those same points. Previously, Jung & Zhang (2018) proved convergence of the one-point marginals to totally skewed $\alpha$-stable distributions for a single symmetric cusp. The limits we prove here are stronger, since they are in the functional sense, but also allow for more varied behaviour due to the presence of multiple cusps. In particular, the general limits we obtain allow for any skewness parameter, as opposed to just the totally skewed cases. We also show that convergence in the stronger $J_1$-topology is not possible.