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 [Zha17] and [JZ18]. In particular, we show that properly normalized Birkhoff sums of Hölder observables, with respect to the billiard map, converge in Skorokhod's M 1 -topology to an α-stable Lévy motion, where α depends on the 'curvature' of the flattest points and the skewness parameter ξ depends on the values of the observable at those same points. Previously, [JZ18] proved convergence of the one-point marginals to totally skewed α-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 J 1 -topology is not possible.

Necessary and sufficient condition for ℳ2-convergence to a Lévy process for billiards with cusps at flat points

“…Recently, Zhang [10] introduced a class of billiards with cusps where the boundary has vanishing curvature at a cusp. For definiteness, we suppose throughout that there is a single symmetric cusp; more general examples are considered in [2]. Jung & Zhang [3] proved convergence to an α-stable law (with normalization n 1/α ) for such billiards; any α ∈ (1, 2) can be achieved depending on the flatness at the cusp.…”

“…Jung & Zhang [3] proved convergence to an α-stable law (with normalization n 1/α ) for such billiards; any α ∈ (1, 2) can be achieved depending on the flatness at the cusp. Then in two papers [2,4] written independently and using different methods, we obtained functional versions, yielding weak convergence to the corresponding α-stable Lévy process.…”

“…For Hölder (and hence bounded) observables, it is easy to see that convergence in the standard J 1 Skorohod topology [7] fails. Convergence in the M 1 Skorohod topology is proved in [2] for observables that have constant sign near the cusp. In [4], necessary and sufficient conditions for M 1 -convergence, and sufficient conditions for convergence in the M 2 topology, are given.…”

“…Moreover, it is conjectured in [4] that the sufficient conditions for M 2 -convergence are necessary and hence that there are observables for which convergence fails in all of the Skorohod topologies [7]. (For more details about the various Skorohod topologies, we refer to [2,4,5,7,9]. )…”

Convergence to a Lévy Process in the Skorohod $${{\mathcal {M}}}_1$$ and $${{\mathcal {M}}}_2$$ Topologies for Nonuniformly Hyperbolic Systems, Including Billiards with Cusps

“…(a) WIP holds in the M 1 (hence also in the M 2 ) topology; (b) WIP holds in the M 2 topology but not in the M 1 topology; (c) the WIP does not hold in the M 1 topology, and we conjecture that the WIP does not hold even in the M 2 topology. Remark 1.5 After writing this paper, we learned of independent work of [26] for a related billiard example but with three cusps at flat points. They considered the case where v has constant sign near each cusp and proved convergence to a Lévy process in the M 1 topology.…”