Stable Laws for Chaotic Billiards with Cusps at Flat Points

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 Hölder continuous observables, we show that properly normalized Birkhoff sums, with respect to the billiard map, converge in law to a totally skewed α-stable law.

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Acknowledgements

The research of H. Zhang was supported in part by NSF Grant DMS-1151762 and also in part by a grant from the Simons Foundation (337646, HZ). The research of P. Jung was supported in part by NSA Grant H98230-14-1-0144 and NRF Grant N01170220. We would like to thank Dmitry Dolgopyat for posing the questions and also suggesting the main results discussed in this paper, i.e., the emergence of stable laws in billiard systems exhibiting slow decay of correlations. H. Zhang also thanks him for many invaluable discussions and suggestions.

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Correspondence to Paul Jung.

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Communicated by Dmitry Dolgopyat.

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Jung, P., Zhang, HK. Stable Laws for Chaotic Billiards with Cusps at Flat Points. Ann. Henri Poincaré 19, 3815–3853 (2018). https://doi.org/10.1007/s00023-018-0726-y

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