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Statistical properties of the maximal entropy measure for partially hyperbolic attractors

Published online by Cambridge University Press:  28 January 2016

ARMANDO CASTRO  and
TEÓFILO NASCIMENTO
ARMANDO CASTRO
Affiliation:
Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110, Salvador-Ba, Brazil email armando@impa.br
TEÓFILO NASCIMENTO
Affiliation:
Departamento de Ciências Exatas e da Terra – Campus II, Universidade do Estado da Bahia, Br 110, km 03, 48.040-210, Alagoinhas-Ba, Brazil email atnascimento@uneb.br
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Abstract

We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semiconjugate to non-uniformly expanding maps. Using the theory of projective metrics on cones, we then prove exponential decay of correlations for Hölder continuous observables and the central limit theorem for the maximal entropy probability measure. Moreover, for systems derived from a solenoid, we also prove the statistical stability for the maximal entropy probability measure. Finally, we use such techniques to obtain similar results in a context containing partially hyperbolic systems derived from Anosov.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 4 , June 2017 , pp. 1060 - 1101
Copyright
© Cambridge University Press, 2016 

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References

Alves, J. F., Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351–398.Google Scholar
Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Séries in Nonlinear Dynamics, 16) . World Scientific, London, 2000.CrossRefGoogle Scholar
Baladi, V.. Anisotropic Sobolev spaces and dynamical transfer operators: C ∞ foliations. Algebraic and Topological Dynamics (Contemporary Mathematics) . Eds. Kolyada, S., Manin, Y. and Ward, T.. American Mathematical Society, New York, 2005, pp. 123–136.CrossRefGoogle Scholar
Buzzi, J. and Fisher, T.. Intrinsic ergodicity for certain nonhyperbolic robustly transitive systems. Preprint, 2009, arXiv:0903.3692.Google Scholar
Buzzi, J. and Fisher, T.. Entropic stability beyond partial hyperbolicity. J. Mod. Dyn. 7(4) (2013), 527–552.CrossRefGoogle Scholar
Buzzi, J., Fisher, T., Sambarino, M. and Vásquez, C.. Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems. Ergod. Th. & Dynam. Sys. 32(01) (2012), 63–79.Google Scholar
Baladi, V. and Gouezel, S. S.. Banach spaces for piecewise cone hyperbolic maps. J. Mod. Dyn. 4 (2010), 91–137.CrossRefGoogle Scholar
Bruin, H. and Keller, G.. Equilibrium states for S-unimodal maps. Ergod. Th. & Dynam. Sys. 18 (1998), 765–789.Google Scholar
Blank, M., Keller, G. and Liverani, C.. Ruelle–Perron–Frobenius spectrum for Anosov maps. Nonlinearity 15 (2001), 1905–1973.CrossRefGoogle Scholar
Baladi, V. and Liverani, C.. Exponential decay of correlations for piecewise cone hyperbolic contact flows. Comm. Math. Phys. 314 (2012), 689–773.CrossRefGoogle Scholar
Buzzi, J. and Maume-Deschamps, V.. Decay of correlations for piecewise invertible maps in higher dimensions. Israel J. Math. 131 (2002), 203–220.CrossRefGoogle Scholar
Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470) . Springer, New York, 1975.Google Scholar
Bowen, R. and Ruelle, D.. The ergodic theory of Axiom A flows. Invent. Math. 29 (1975), 181–202.Google Scholar
Buzzi, J. and Sarig, O.. Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. Ergod. Th. & Dynam. Sys. 23 (2003), 1383–1400.Google Scholar
Baladi, V. and Tsujii, M.. Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Ann. Inst. Fourier 57 (2007), 27–54.Google Scholar
Bruin, H. and Todd, M.. Equilibrium states for interval maps: potentials with sup𝜑-inf𝜑 < h top(f). Comm. Math. Phys. 283 (2008), 579–611.Google Scholar
Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. (1999).Google Scholar
Carvalho, M.. Sinai–Ruelle–Bowen measures for N-dimensional derived from Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 13 (1993), 21–44.Google Scholar
Castro, A.. Backward inducing and exponential decay of correlations for partially hyperbolic attractors with mostly contracting central directions. PhD Thesis, IMPA, 1998.Google Scholar
Castro, A.. Backward inducing and exponential decay of correlations for partially hyperbolic attractors. Israel J. Math. 130 (2002), 29–75.Google Scholar
Castro, A.. Fast mixing for attractors with mostly contracting central direction. Ergod. Th. & Dynam. Sys. 24 (2004), 17–44.Google Scholar
Climenhaga, V., Fisher, T. and Thompson, D.. Unique equilibrium states for the robustly transitive diffeomorphisms of Mañé and Bonatti–Viana. Preprint, 2015, arXiv:1505.06371.Google Scholar
Castro, A. and Varandas, P.. Equilibrium states for non-uniformly expanding maps: decay of correlations and strong stability. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(2) (2013), 225–249.Google Scholar
Demers, M. and Liverani, C.. Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc. 360 (2008), 4777–4814.Google Scholar
Gouezel, S. and Liverani, C.. Banach spaces adapted to Anosov systems. Ergod. Th. & Dynam. Sys. 26 (2006), 189–217.Google Scholar
Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583) . Springer, Berlin, 1977.Google Scholar
Liverani, C.. Decay of correlations. Ann. of Math. (2) 142 (1995), 239–301.Google Scholar
Luzzatto, S. and Melbourne, I.. Statistical properties and decay of correlations for interval maps with critical points and singularities. Comm. Math. Phys. 320 (2013), 21–35.CrossRefGoogle Scholar
Liverani, C., Saussol, B. and Vaienti, S.. Conformal measure and decay of correlation for covering weighted systems. Ergod. Th. & Dynam. Sys. 18(6) (1998), 1399–1420.Google Scholar
Liverani, C. and Terhesiu, D.. Mixing for some non-uniformly hyperbolic systems. Ann. Henri Poincaré, to appear, Preprint, 2013, arXiv:1308.2422v2.Google Scholar
Mañé, R.. Introdução à Teoria Ergódica Coleção Projeto Euclides (Instituto de Matemática Pura e Aplicada), Livros Técnicos e Científicos Editora, 1983.Google Scholar
Mañé, R.. Contributions to the stability conjecture. Topology 17(4) (1978), 383–396.Google Scholar
Melbourne, I.. Mixing for invertible dynamical systems with infinite measure. Stoch. Dyn. 15 (2015).Google Scholar
Melbourne, I. and Terhesiu, D.. Decay of correlations for nonuniformly expanding systems with general return times. Ergod. Th. & Dynam. Sys. 34 (2014), 893–918.Google Scholar
Oliveira, K. and Viana, M.. Thermodynamical formalism for an open classes of potentials and non-uniformly hyperbolic maps. Ergod. Th. & Dynam. Sys. 28 (2008).Google Scholar
Pinheiro, V.. Expanding measures. Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), 889–939.Google Scholar
Sarig, O.. Thermodynamic formalism for countable Markov shifts. Ergod. Th. & Dynam. Sys. 19 (1999), 1565–1593.Google Scholar
Sinai, Ya.. Gibbs measures in ergodic theory. Russian Math. Surveys 27 (1972), 21–69.Google Scholar
Sambarino, M. and Vásquez, C.. Bowen measure for derived from Anosov diffeomorphims. Preprint, 2009, arXiv:0904.1036.Google Scholar
Ures, R.. Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part. Proc. Amer. Math. Soc. 140 (2012).Google Scholar
Viana, M.. Stochastic dynamics of deterministic systems. Colóq. Bras. Mat. (1997).Google Scholar
Varandas, P. and Viana, M.. Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 555–593.Google Scholar
Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.Google Scholar
Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147(3) (1998), 585D-650.CrossRefGoogle Scholar
Yuri, M.. Thermodynamical formalism for countable to one Markov systems. Trans. Amer. Math. Soc. 335 (2003), 2949–2971.Google Scholar