Abstract
We prove uniqueness of equilibrium states for a family of partially hyperbolic horseshoes associated to a class of Hölder continuous potentials with small variation and derive statistical properties for this unique equilibrium. We define a projection map associated to the horseshoe and prove a spectral gap for its transfer operator acting on some space of Hölder continuous observables. From this we deduce an exponential decay of correlations and a central limit theorem. We finally extend these results to the horseshoe via Rohlin’s disintegration of the equilibrium along the stable fibers.
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References
Arbieto, A., Prudente, L.: Uniqueness of equilibrium states for some partially hyperbolic horseshoes. Discrete Contin. Dyn. Syst. 32(1), 27–40 (2012)
Baladi, V.: Positive Transfer Operators and Decay of Correlations. World Scientific Publishing, Singapore (2000)
Birkhoff, G.: Lattice Theory American Mathematical Society Colloquium Publications 25, 4th edn. American Mathematical Society, Providence (1979)
Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc. 153, 401–414 (1971)
Buzzing, J., Fisher, T., Sambarino, M., Vásquez, C.: Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems. Ergodic Theory Dyn. Syst. 32, 63–79 (2011)
Buzzi, J., Pacault, F., Schmitt, B.: Conformal measures for multidimensional piecewise invertible maps. Ergodic Theory Dyn. Syst. 21, 1035–1049 (2001)
Buzzi, J., Sarig, O.: Uniqueness of equilibrium measures for countable markov shifts. Ergodic Theory Dyn. Syst. 23, 1383–1400 (2003)
Castro, A., Nascimento, T.: Statistical properties of the maximal entropy measure for partially hyperbolic attractors. Ergodic Theory Dyn. Syst. Available on CJO2016
Castro, A., Varandas, P.: Equilibrium states for non-uniformly expanding maps: decay of correlations and strong stability. Annalles de l’Institut Henri Poincaré-Analyse Non-Linéaire, 225–249 (2013)
Díaz, L., Horita, V., Rios, I., Sambarino, M.: Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergodic Theory Dyn. Syst. 29, 433–474 (2009)
Dürr, D., Goldstein, S.: Remarks on the central limit theorem for weakly dependence random variables. Stochastic processes—mathematics and physics, Lect. Notes in Math., vol. 1158, pp. 104–118. Springer, Berlin (1986)
Ferrero, P., Schmitt, B.: Ruelle’s Perron–Frobenius theorem and projective metrics Coll. Math. Soc. János Bolyai 27, 333–336 (1979)
Keller, G.: Un theorème de la limite centrale pour une classe de tranformations monotones par morceaux. C.R.A.S. A 291, pp. 155–158 (1980)
Keller, G., Nowicki, T.: Spectral theory, zeta functions and the distribution of periodic points for Collet–Eckmann maps Comm. Math. Phys. 149, 31–69 (1992)
Leplaideur, R., Oliveira, K., Rios, I.: Equilibrium States for partially hyperbolic horseshoes. Ergodic Theory Dyn. Syst. 31, 179–195 (2011)
Liverani, C.: Decay of correlations. Ann. Math. 142, 239–301 (1995)
Liverani, C.: Decay of correlations for piecewise expanding maps. J. Stat. Phys. 78, 1111–1129 (1995)
Oliveira, K., Viana, M.: Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps. Ergodic Theory Dyn. Syst. 28, 501–533 (2008)
Rios, I., Siqueira, J.: On equilibrium states for partially hyperbolic horseshoes. Ergodic Theory Dyn. Syst. (2016) (to appear)
Rohlin, V.A.: On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, vol. 71, p. 55 (1952)
Ruelle, D.: Statistical mechanics of a one-dimensional lattice gas. Commun. Math. Phys. 9(4), 267–278 (1968)
Ruelle, D.: Thermodynamic formalism, Encyclopedia Mathematics and its Applications, vol. 5. Addison-Wesley Publishing Company (1978)
Sarig, O.: Thermodynamic formalism for countable Markov shifts. Ergodic Theory Dyn. Syst. 19, 1565–1593 (1999)
Sarig, O.: Existence of Gibbs measures for countable Markov shifts. Proc. Am. Math. Soc. 131, 751–758 (2003) (Electronic)
Sinai, Y.: Gibbs measures in ergodic theory. Russ. Math. Surveys 27, 21–69 (1972)
Simmons, D.: Conditional measures and conditional expectation; Rohlin’s disintegration theorem. Discrete Contin. Dyn. Syst. 32(7), 2565–2582 (2012)
Varandas, P., Viana, M.: Existence, uniqueness and stability of equilibrium states for non-hyperbolic expanding maps, Ann. I. H. Poincaré, AN 27, pp. 555–593 (2010)
Viana, M.: Stochastic dynamics of deterministic systems, Colóquio Brasileiro de Matemática (1997)
Young, L.S.: Decay of correlations for certain quadratic maps. Commun. Math. Phys. 146, 123–138 (1992)
Acknowledgements
This work was carried out at Universidade do Porto. The authors are very thankful to Silvius Klein for the help with the manuscript version and encouragement. VR is grateful to Ivaldo Nunes for the encouragement.
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The authors were supported by CNPq-Brazil.
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Ramos, V., Siqueira, J. On Equilibrium States for Partially Hyperbolic Horseshoes: Uniqueness and Statistical Properties. Bull Braz Math Soc, New Series 48, 347–375 (2017). https://doi.org/10.1007/s00574-017-0027-y
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DOI: https://doi.org/10.1007/s00574-017-0027-y