SRB Measures for Polygonal Billiards with Contracting Reflection Laws

We study polygonal billiards with reflection laws contracting the angle of reflection towards the normal. It is shown that if a polygon does not have parallel sides facing each other, then the corresponding billiard map has finitely many ergodic Sinai–Ruelle–Bowen measures whose basins cover a set of full Lebesgue measure.

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Teichmüller dynamics and unique ergodicity via currents and Hodge theory

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2019-0037 ◽

2020 ◽

Vol 2020 (768) ◽

pp. 39-54

Author(s):

Curtis T. McMullen

Keyword(s):

Hodge Theory ◽

Unique Ergodicity ◽

Polygonal Billiards ◽

Teichmuller Dynamics

AbstractWe present a cohomological proof that recurrence of suitable Teichmüller geodesics implies unique ergodicity of their terminal foliations. This approach also yields concrete estimates for periodic foliations and new results for polygonal billiards.

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The set of points with Markovian symbolic dynamics for non-uniformly hyperbolic diffeomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.114 ◽

2020 ◽

pp. 1-26

Author(s):

SNIR BEN OVADIA

Keyword(s):

Symbolic Dynamics ◽

Full Measure ◽

Probability Measures ◽

Positive Entropy ◽

Surface Diffeomorphisms ◽

Srb Measures ◽

Hyperbolic Diffeomorphisms ◽

Set Of Points ◽

Math Phys ◽

Homoclinic Classes

Abstract The papers [O. M. Sarig. Symbolic dynamics for surface diffeomorphisms with positive entropy. J. Amer. Math. Soc.26(2) (2013), 341–426] and [S. Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. J. Mod. Dyn.13 (2018), 43–113] constructed symbolic dynamics for the restriction of $C^r$ diffeomorphisms to a set $M'$ with full measure for all sufficiently hyperbolic ergodic invariant probability measures, but the set $M'$ was not identified there. We improve the construction in a way that enables $M'$ to be identified explicitly. One application is the coding of infinite conservative measures on the homoclinic classes of Rodriguez-Hertz et al. [Uniqueness of SRB measures for transitive diffeomorphisms on surfaces. Comm. Math. Phys.306(1) (2011), 35–49].

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An analytical construction of the SRB measures for Baker-type maps

Chaos An Interdisciplinary Journal of Nonlinear Science ◽

10.1063/1.166324 ◽

1998 ◽

Vol 8 (2) ◽

pp. 424-443 ◽

Cited By ~ 22

Author(s):

S. Tasaki ◽

Thomas Gilbert ◽

J. R. Dorfman

Keyword(s):

Srb Measures ◽

Analytical Construction

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SRB measures for attractors with continuous invariant splittings

Mathematische Zeitschrift ◽

10.1007/s00209-017-1883-2 ◽

2017 ◽

Vol 288 (1-2) ◽

pp. 135-165 ◽

Cited By ~ 1

Author(s):

Zeya Mi ◽

Yongluo Cao ◽

Dawei Yang

Keyword(s):

Srb Measures

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SRB measures for partially hyperbolic attractors of local diffeomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.115 ◽

2018 ◽

Vol 40 (6) ◽

pp. 1545-1593

Author(s):

ANDERSON CRUZ ◽

PAULO VARANDAS

Keyword(s):

Thermodynamic Formalism ◽

Positive Lebesgue Measure ◽

Stable Bundle ◽

Local Diffeomorphism ◽

Local Diffeomorphisms ◽

Axiom A ◽

Srb Measures ◽

Cone Field ◽

Hyperbolic Attractors ◽

Partially Hyperbolic

We contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures are statistically stable (in the weak$^{\ast }$ topology) and that their entropy varies continuously with respect to the local diffeomorphism.

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