Bernoulli Property for Some Hyperbolic Billiards

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Abstract

We prove that hyperbolic billiards constructed by Bussolari and Lenci are Bernoulli systems. These billiards cannot be studied by existing approaches to analysis of billiards that have some focusing boundary components, which require the diameter of the billiard table to be of the same order as the largest curvature radius along the focusing component. Our proof employs a local ergodic theorem which states that, under certain conditions, there is a full measure set of the billiard phase space such that each point of the set has a neighborhood contained (mod 0) in a Bernoulli component of the billiard map.

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ACKNOWLEDGMENTS

This work is the result of my PhD thesis under the supervision of Roberto Markarian who welcomed me during an internship at the IMERL, Universidad de la República, Uruguay. I also give special thanks to Rodrigo Bissacot, my advisor at the Universidade de São Paulo, Brasil.

Funding

My research is supported by CAPES Grant 8123/13-6.

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Correspondence to Rodrigo M.D. Andrade.

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MSC2010

37D50, 37D25

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Andrade, R.M. Bernoulli Property for Some Hyperbolic Billiards. Regul. Chaot. Dyn. 25, 349–382 (2020). https://doi.org/10.1134/S1560354720040048

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Keywords

  • hyperbolic billiards
  • Bernoulli property
  • focusing billiards