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On the Bernoulli Property of Planar Hyperbolic Billiards

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Abstract

We consider billiards in non-polygonal domains of the plane with boundary consisting of curves of three different types: straight segments, strictly convex inward curves and strictly convex outward curves of a special kind. The billiard map for these domains is known to have non-vanishing Lyapunov exponents a.e. provided that the distance between the curved components of the boundary is sufficiently large, and the set of orbits having collisions only with the flat part of the boundary has zero measure. Under a few additional conditions, we prove that there exists a full measure set of the billiard phase space such that each of its points has a neighborhood contained up to a zero measure set in one Bernoulli component of the billiard map. Using this result, we show that there exists a large class of planar hyperbolic billiards that have the Bernoulli property. This class includes the billiards in convex domains bounded by straight segments and strictly convex inward arcs constructed by Donnay.

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal da Bahia, Avenida Adhemar de Barros, Ondina, Salvador, BA, 40170110, Brazil

    Gianluigi Del Magno

  2. Instituto de Matemática y Estadística ‘Prof. Ing. Rafael Laguardia’ (IMERL), Facultad de Ingeniería, Universidad de la República, Montevideo, Uruguay

    Roberto Markarian

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  1. Gianluigi Del Magno
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  2. Roberto Markarian
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Correspondence to Gianluigi Del Magno.

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Communicated by C. Liverani

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Del Magno, G., Markarian, R. On the Bernoulli Property of Planar Hyperbolic Billiards. Commun. Math. Phys. 350, 917–955 (2017). https://doi.org/10.1007/s00220-017-2828-7

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