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 Regul. Chaotic Dyn., 2013, Volume 18, Issue 4, Pages 425–452 (Mi rcd121)

Singular Sets of Planar Hyperbolic Billiards are Regular

Gianluigi Del Magnoa, Roberto Markarianb

a CEMAPRE, ISEG, Universidade Técnica de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal
b Instituto de Matemática y Estadística “Prof. Ing. Rafael Laguardia” (IMERL), Facultad de Ingeniería, Universidad de la República, Montevideo, Uruguay

Abstract: Many planar hyperbolic billiards are conjectured to be ergodic. This paper represents a first step towards the proof of this conjecture. The Hopf argument is a standard technique for proving the ergodicity of a smooth hyperbolic system. Under additional hypotheses, this technique also applies to certain hyperbolic systems with singularities, including hyperbolic billiards. The supplementary hypotheses concern the subset of the phase space where the system fails to be $C^2$ differentiable. In this work, we give a detailed proof of one of these hypotheses for a large collection of planar hyperbolic billiards. Namely, we prove that the singular set and each of its iterations consist of a finite number of compact curves of class $C^2$ with finitely many intersection points.

Keywords: hyperbolic billiards, ergodicity

 Funding Agency Grant Number Fundação para a Ciência e a Tecnologia PTDC-MAT-105448-2008 G. Del Magno was supported by Fundação para a Ciência e a Tecnologia through the Program POCI 2010 and the Project “Randomness in Deterministic Dynamical Systems and Applications” (PTDC-MAT-105448-2008).

DOI: https://doi.org/10.1134/S1560354713040072

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MSC: 37D50, 37A25, 37D25, 37N05
Accepted:07.05.2013
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Citation: Gianluigi Del Magno, Roberto Markarian, “Singular Sets of Planar Hyperbolic Billiards are Regular”, Regul. Chaotic Dyn., 18:4 (2013), 425–452

Citation in format AMSBIB
\Bibitem{DelMar13} \by Gianluigi Del Magno, Roberto Markarian \paper Singular Sets of Planar Hyperbolic Billiards are Regular \jour Regul. Chaotic Dyn. \yr 2013 \vol 18 \issue 4 \pages 425--452 \mathnet{http://mi.mathnet.ru/rcd121} \crossref{https://doi.org/10.1134/S1560354713040072} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3090210} \zmath{https://zbmath.org/?q=an:1338.37049} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000322878100007} 

• http://mi.mathnet.ru/eng/rcd121
• http://mi.mathnet.ru/eng/rcd/v18/i4/p425

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. G. Del Magno, R. Markarian, “On the Bernoulli property of planar hyperbolic billiards”, Commun. Math. Phys., 350:3 (2017), 917–955