Sinai’s Work on Markov Partitions and SRB Measures

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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Construction of Markov Partitions in PL1D Maps

IEEE Transactions on Circuits & Systems II Express Briefs ◽

10.1109/tcsii.2013.2278106 ◽

2013 ◽

Vol 60 (10) ◽

pp. 702-706 ◽

Cited By ~ 3

Author(s):

Toni Draganov Stojanovski ◽

Ljupco Kocarev

Keyword(s):

Markov Partitions

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Symbolic dynamics and Markov partitions

Bulletin of the American Mathematical Society ◽

10.1090/s0273-0979-98-00737-x ◽

1998 ◽

Vol 35 (01) ◽

pp. 1-57 ◽

Cited By ~ 41

Author(s):

Roy L. Adler

Keyword(s):

Symbolic Dynamics ◽

Markov Partitions

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Symbolic dynamics for non-uniformly hyperbolic systems

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.80 ◽

2020 ◽

pp. 1-68

Author(s):

YURI LIMA

Keyword(s):

Symbolic Dynamics ◽

Hyperbolic Systems ◽

Markov Partition ◽

Equilibrium Measures ◽

Markov Partitions ◽

Recent Advances ◽

Closed Orbits ◽

Symbolic Extension ◽

Uniformly Hyperbolic Systems

Abstract This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

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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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