Symbolic dynamics for non-uniformly  hyperbolic surface maps  with discontinuities

Lima, Yuri; Matheus, Carlos

doi:10.24033/asens.2350

Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities
[Dynamique symbolique pour les systèmes non-uniformément hyperboliques avec discontinuités]

Lima, Yuri ; Matheus, Carlos

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 1, pp. 1-38.

Résumé
Abstract

Nous construisons une dynamique symbolique pour les applications non uniformément hyperboliques d'une surface ayant un ensemble de discontinuités $𝒟$ . La dérivée de l'application peut ne pas être bornée, de l'ordre d'une puissance négative de la distance à $𝒟$ . Sous certaines conditions géométriques naturelles sur l'espace des phases $M$ , nous codifions un ensemble d'orbites non uniformément hyperboliques qui ne s'approchent pas exponentiellement vite de $𝒟$ . Notre résultat s'applique aux billards planaires non uniformément hyperboliques tels que les billards de Bunimovich.

This work constructs symbolic dynamics for non-uniformly hyperbolic surface maps with a set of discontinuities $𝒟$ . We allow the derivative of points nearby $𝒟$ to be unbounded, of the order of a negative power of the distance to $𝒟$ . Under natural geometrical assumptions on the underlying space $M$ , we code a set of non-uniformly hyperbolic orbits that do not converge exponentially fast to $𝒟$ . The results apply to non-uniformly hyperbolic planar billiards, e.g., Bunimovich billiards.

MR Zbl

DOI : 10.24033/asens.2350

Classification : 37B10, 37D25, 37D50; 37C35.
Keywords: Billiards, Markov partition, symbolic dynamics.
Mot clés : Billards, Partition de Markov, dynamique symbolique.

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     author = {Lima, Yuri and Matheus, Carlos},
     title = {Symbolic dynamics for non-uniformly  hyperbolic surface maps  with discontinuities},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1--38},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {1},
     year = {2018},
     doi = {10.24033/asens.2350},
     mrnumber = {3764037},
     zbl = {1444.37011},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2350/}
}

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Lima, Yuri; Matheus, Carlos. Symbolic dynamics for non-uniformly  hyperbolic surface maps  with discontinuities. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 1, pp. 1-38. doi : 10.24033/asens.2350. http://www.numdam.org/articles/10.24033/asens.2350/

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