Hyperbolic polygonal billiards with finitely many ergodic SRB measures

Magno, Gianluigi Del; Dias, João Lopes; Duarte, Pedro; Gaivão, José Pedro

doi:10.1017/etds.2016.119

Cited by 5 publications

(2 citation statements)

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“…It is known that strongly contracting billiard maps on generic convex polygons are uniformly hyperbolic and have finite number of ergodic SRB measures [5]. Recently, it has been proved that the same conclusion hods for contracting billiard maps on polygons with no parallel sides facing each other (even for contracting reflection laws close to the specular and for non-convex polygons) [7].…”

Section: Introductionmentioning

confidence: 87%

Hyperbolic billiards on polytopes with contracting reflection laws

Duarte¹,

Gaivão²,

Soufi³

2017

Discrete &Amp; Continuous Dynamical Systems - A

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We study billiards on polytopes in R d with contracting reflection laws, i.e. non-standard reflection laws that contract the reflection angle towards the normal. We prove that billiards on generic polytopes are uniformly hyperbolic provided there exists a positive integer k such that for any k consecutive collisions, the corresponding normals of the faces of the polytope where the collisions took place generate R d . As an application of our main result we prove that billiards on generic polytopes are uniformly hyperbolic if either the contracting reflection law is sufficiently close to the specular or the polytope is obtuse. Finally, we study in detail the billiard on a family of 3-dimensional simplexes.

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Section: Introductionmentioning

confidence: 87%

Hyperbolic billiards on polytopes with contracting reflection laws

Duarte¹,

Gaivão²,

Soufi³

2017

Discrete &Amp; Continuous Dynamical Systems - A

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“…If the billiard in P has no such orbits, then for any λ ∈ (0, 1), the pinball billiard in P has a finite number of ergodic Sinai-Ruelle-Bowen (SRB) measures, each supporting a uniformly hyperbolic attractor. The union of the basins of attraction of the SRB measures has full Lebesgue measure [7], and by Pesin theory, the periodic orbits are dense in the support of the SRB measures.…”

Section: Introductionmentioning

confidence: 99%

λ-stability of periodic billiard orbits

Gaivão

Troubetzkoy

2018

Nonlinearity

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Abstract. We introduce a new notion of stability for periodic orbits in polygonal billiards. We say that a periodic orbit of a polygonal billiard is λ-stable if there is a periodic orbit for the corresponding pinball billiard which converges to it as λ → 1. This notion of stability is unrelated to the notion introduced by Galperin, Stepin and Vorobets. We give sufficient and necessary conditions for a periodic orbit to be λ-stable and prove that the set of d-gons having at most finite number of λ-stable periodic orbits is dense is the space of d-gons. Moreover, we also determine completely the λ-stable periodic orbits in integrable polygons.

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