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February  2020, 40(2): 767-780. doi: 10.3934/dcds.2020061

Equilibrium states of almost Anosov diffeomorphisms

Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

Received  January 2019 Revised  August 2019 Published  November 2019

We develop a thermodynamic formalism for a class of diffeomorphisms of a torus that are "almost-Anosov". In particular, we use a Young tower construction to prove the existence and uniqueness of equilibrium states for a collection of non-Hölder continuous geometric potentials over almost Anosov systems with an indifferent fixed point, as well as prove exponential decay of correlations and the central limit theorem for these equilibrium measures.

Citation: Dominic Veconi. Equilibrium states of almost Anosov diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 767-780. doi: 10.3934/dcds.2020061
References:
[1]

J. Alves and R. Leplaideur, SRB measures for almost axiom A diffeomorphisms, Ergod. Th. & Dynam. Sys., 36 (2016), 2015-2043.  doi: 10.1017/etds.2015.4.  Google Scholar

[2]

A. Gura, (Russian) [http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=7623&option_lang=eng Separating diffeomorphisms of the torus], Mat. Zamekti, 18 (1975), 41–49. Google Scholar

[3]

H. Hu, Conditions for the existence of SBR measures of "almost Anosov" diffeomorphisms, Trans. Amer. Math. Soc., 352 (2000), 2331-2367.  doi: 10.1090/S0002-9947-99-02477-0.  Google Scholar

[4]

H. Hu and L. S. Young, Nonexistence of SBR measures for some diffeomorphisms that are "almost Anosov", Ergod. Th. & Dynam. Sys., 15 (1995), 67-76.  doi: 10.1017/S0143385700008245.  Google Scholar

[5]

A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2), 110 (1979), 529–547. doi: 10.2307/1971237.  Google Scholar

[6]

F. Ledrappier and L. S. Young, The metric entropy of diffeomorphisms, Bull. of the Amer. Math. Soc. (N.S.), 11 (1984), 343-346.  doi: 10.1090/S0273-0979-1984-15299-6.  Google Scholar

[7]

Y. PesinS. Senti and K. Zhang, Thermodynamics of the Katok map, Ergodic Theory Dynam. Systems, 39 (2019), 764-794.  doi: 10.1017/etds.2017.35.  Google Scholar

[8]

Y. PesinS. Senti and K. Zhang, Thermodynamics of towers of hyperbolic type, Trans. Amer. Math. Soc., 368 (2016), 8519-8552.  doi: 10.1090/tran/6599.  Google Scholar

[9]

F. Rodriguez-HertzM. A. Rodriguez-HertzA. Tahzibi and R. Ures, Uniqueness of SRB measures for transitive diffeomorphisms on surfaces, Commun. Math. Phys., 306 (2011), 35-49.  doi: 10.1007/s00220-011-1275-0.  Google Scholar

show all references

References:
[1]

J. Alves and R. Leplaideur, SRB measures for almost axiom A diffeomorphisms, Ergod. Th. & Dynam. Sys., 36 (2016), 2015-2043.  doi: 10.1017/etds.2015.4.  Google Scholar

[2]

A. Gura, (Russian) [http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=7623&option_lang=eng Separating diffeomorphisms of the torus], Mat. Zamekti, 18 (1975), 41–49. Google Scholar

[3]

H. Hu, Conditions for the existence of SBR measures of "almost Anosov" diffeomorphisms, Trans. Amer. Math. Soc., 352 (2000), 2331-2367.  doi: 10.1090/S0002-9947-99-02477-0.  Google Scholar

[4]

H. Hu and L. S. Young, Nonexistence of SBR measures for some diffeomorphisms that are "almost Anosov", Ergod. Th. & Dynam. Sys., 15 (1995), 67-76.  doi: 10.1017/S0143385700008245.  Google Scholar

[5]

A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2), 110 (1979), 529–547. doi: 10.2307/1971237.  Google Scholar

[6]

F. Ledrappier and L. S. Young, The metric entropy of diffeomorphisms, Bull. of the Amer. Math. Soc. (N.S.), 11 (1984), 343-346.  doi: 10.1090/S0273-0979-1984-15299-6.  Google Scholar

[7]

Y. PesinS. Senti and K. Zhang, Thermodynamics of the Katok map, Ergodic Theory Dynam. Systems, 39 (2019), 764-794.  doi: 10.1017/etds.2017.35.  Google Scholar

[8]

Y. PesinS. Senti and K. Zhang, Thermodynamics of towers of hyperbolic type, Trans. Amer. Math. Soc., 368 (2016), 8519-8552.  doi: 10.1090/tran/6599.  Google Scholar

[9]

F. Rodriguez-HertzM. A. Rodriguez-HertzA. Tahzibi and R. Ures, Uniqueness of SRB measures for transitive diffeomorphisms on surfaces, Commun. Math. Phys., 306 (2011), 35-49.  doi: 10.1007/s00220-011-1275-0.  Google Scholar

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