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

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 Dynamical Systems, Ergodic Theory, and Probability: in Memory of Kolya Chernov (Contemporary Mathematics vol 698) (Providence, RI: American Mathematical Society) pp 287–316) and Jung and Zhang (2018 Ann. Henri Poincaré 19 3815–53). In particular, we show that properly normalized Birkhoff sums of Hölder observables, with respect to the billiard map, converge in Skorokhod’s M1-topology to an -stable Lévy motion, where depends on the ‘curvature’ of the flattest points and the skewness parameter depends on the values of the observable at those same points. Previously, Jung and Zhang (2018 Ann. Henri Poincaré 19 3815–53) proved convergence of the one-point marginals to totally skewed -stable distributions for a 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 J1-topology is not possible.