The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors

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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Symbolic dynamics for surface diffeomorphisms with positive entropy

Journal of the American Mathematical Society ◽

10.1090/s0894-0347-2012-00758-9 ◽

2012 ◽

Vol 26 (2) ◽

pp. 341-426 ◽

Cited By ~ 22

Author(s):

Omri M. Sarig

Keyword(s):

Symbolic Dynamics ◽

Positive Entropy ◽

Surface Diffeomorphisms

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On C*-Algebras Associated with Subshifts

International Journal of Mathematics ◽

10.1142/s0129167x97000172 ◽

1997 ◽

Vol 08 (03) ◽

pp. 357-374 ◽

Cited By ~ 37

Author(s):

Kengo Matsumoto

Keyword(s):

Symbolic Dynamics ◽

Markov Shift ◽

C Algebra ◽

C Algebras ◽

Markov Shifts ◽

Universal Properties

We construct and study C*-algebras associated with subshifts in symbolic dynamics as a generalization of Cuntz–Krieger algebras for topological Markov shifts. We prove some universal properties for the C*-algebras and give a criterion for them to be simple and purely infinite. We also present an example of a C*-algebra coming from a subshift which is not conjugate to a Markov shift.

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Equilibrium states and zero temperature limit on topologically transitive countable Markov shifts

Transactions of the American Mathematical Society ◽

10.1090/tran/7291 ◽

2018 ◽

Vol 370 (12) ◽

pp. 8451-8465 ◽

Cited By ~ 2

Author(s):

Ricardo Freire ◽

Victor Vargas

Keyword(s):

Temperature Limit ◽

Equilibrium States ◽

Zero Temperature ◽

Topologically Transitive ◽

Markov Shifts ◽

Countable Markov Shifts

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Irreducible complexity of iterated symmetric bimodal maps

Discrete Dynamics in Nature and Society ◽

10.1155/ddns.2005.69 ◽

2005 ◽

Vol 2005 (1) ◽

pp. 69-85 ◽

Cited By ~ 2

Author(s):

J. P. Lampreia ◽

R. Severino ◽

J. Sousa Ramos

Keyword(s):

Tree Structure ◽

Unimodal Maps ◽

Markov Shifts ◽

Irreducible Complexity ◽

The Family

We introduce a tree structure for the iterates of symmetric bimodal maps and identify a subset which we prove to be isomorphic to the family of unimodal maps. This subset is used as a second factor for a∗-product that we define in the space of bimodal kneading sequences. Finally, we give some properties for this product and study the∗-product induced on the associated Markov shifts.

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