Convergence to $α$-stable Lévy motion for chaotic billiards with several cusps at flat points

Paul Jungand Françoise Pène, Hong-Kun Zhang

Published 2018-09-21Version 1

We consider billiards with several cusps at flat points; the case of a single cusp was studied previously in \cite{Z2016b} and \cite{JZ17}. In particular, we show that properly normalized Birkorff 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, with a skewness parameter which depends on the values of the observable at the different flat points. This extends the main result of \cite{JZ17} which proved convergence of the one-point marginals to totally skewed $\alpha$-stable distributions, when there is a single cusp. We also show that convergence in the stronger $J_1$-topology is not possible.