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On equilibrium states for partially hyperbolic horseshoes

Published online by Cambridge University Press:  04 July 2016

I. RIOS and
J. SIQUEIRA
I. RIOS
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Prof. Marcos Waldemar de Freitas Reis, S/N - Bloco H, 4o Andar, 24210-201 Niterǿi, Brazil email rios@mat.uff.br
J. SIQUEIRA
Affiliation:
Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal email jaqueline.rocha@fc.up.pt
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Abstract

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We prove the existence and uniqueness of equilibrium states for a family of partially hyperbolic systems, with respect to Hölder continuous potentials with small variation. The family comes from the projection, on the center-unstable direction, of a family of partially hyperbolic horseshoes introduced by Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys.29 (2009), 433–474]. For the original three-dimensional system we consider potentials with small variation, constant on local stable manifolds, obtaining existence and uniqueness of equilibrium states.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 1 , February 2018, pp. 301 - 335
Copyright
© Cambridge University Press, 2016 

References

Alves, J. F.. SRB measures for non-hyperbolic systems with multidimensional expansion. Ann. Sci. Éc. Norm. Supér. (4) 33(1) (2000), 1–32.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Arbieto, A. and Prudente, L.. Uniqueness of equilibrium states for some partially hyperbolic horseshoes. Discrete Contin. Dyn. Syst. 32(1) (2012), 27–40.CrossRefGoogle Scholar
Bartle, R. G.. The Elements of Real Analysis. Wiley, New York, 1964.Google Scholar
Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414.CrossRefGoogle Scholar
Bowen, R.. Equilibrium States and Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470) . Springer, New York, 1975.CrossRefGoogle Scholar
Buzzi, J., Pacault, F. and Schmitt, B.. Conformal measures for multidimensional piecewise invertible maps. Ergod. Th. & Dynam. Sys. 21 (2001), 1035–1049.CrossRefGoogle Scholar
Buzzi, J. and Sarig, O.. Uniqueness of equilibrium measures for countable markov shifts. Ergod. Th. & Dynam. Sys. 23 (2003), 1383–1400.CrossRefGoogle Scholar
Castro, A. and Nascimento, T.. Statistical properties of the maximal entropy measure for partially hyperbolic attractors. Ergod. Th. & Dynam. Sys. Published online 28 January 2016, doi:10.1017/etds.2015.86.Google Scholar
Denker, M. and Urbanski, M.. On the existence of conformal measures. Trans. Amer. Math. Soc. 328 (1991), 563–587.CrossRefGoogle Scholar
Díaz, L. and Gelfert, K.. Porcupine-like horseshoes: transitivity, Lyapunov Spectrum, and phase transitions. Fund. Math. 216 (2012), 55–100.CrossRefGoogle Scholar
Díaz, L., Horita, V., Rios, I. and Sambarino, M.. Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys. 29 (2009), 433–474.CrossRefGoogle Scholar
Kreyszig, E.. Introductory Functional Analysis with Maps. Wiley, New York, 1989.Google Scholar
Ledrappier, F. and Walters, P.. A relativised variational principle for continuous transformations. J. Lond. Math. Soc. (2) 16 (1977), 568–579.CrossRefGoogle Scholar
Leplaideur, R., Oliveira, K. and Rios, I.. Equilibrium States for partially hyperbolic horseshoes. Ergod. Th. & Dynam. Sys. 31 (2011), 179–195.CrossRefGoogle Scholar
Oliveira, K. and Viana, M.. Existence and uniqueness of maximizing measures for robust classes of local difeomorphisms. Discrete Contin. Dyn. Syst. 15 (2006), 225–236.CrossRefGoogle Scholar
Oliveira, K. and Viana, M.. Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps. Ergod. Th. & Dynam. Sys. 28 (2008), 501–533.CrossRefGoogle Scholar
Pesin, Ya.. Contemporary views and applications. Dimension Theory in Dynamical Systems. University of Chicago Press, Chicago, IL, 1997.CrossRefGoogle Scholar
Pliss, V.. On a conjecture due to Smale. Differ. Uravn. 8 (1972), 262–268.Google Scholar
Ruelle, D.. Statistical mechanics of a one-dimensional lattice gas. Comm. Math. Phys. 9(4) (1968), 267–278.CrossRefGoogle Scholar
Ruelle, D.. Thermodynamic Formalism (Encyclopedia Mathematics and its Applications, 5) . Addison-Wesley Publishing Company, Reading, MA, 1978.Google Scholar
Sarig, O.. Thermodynamic formalism for countable Markov shifts. Ergod. Th. & Dynam. Sys. 19 (1999), 1565–1593.CrossRefGoogle Scholar
Sarig, O.. Existence of Gibbs measures for countable Markov shifts. Proc. Amer. Math. Soc. 131 (2003), 1751–1758 (electronic).CrossRefGoogle Scholar
Sinai, Y.. Gibbs measures in ergodic theory. Russian Math. Surveys 27 (1972), 21–69.CrossRefGoogle Scholar
Varandas, P. and Viana, M.. Existence, uniqueness and stability of equilibrium states for non-hyperbolic expanding maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 555–593.CrossRefGoogle Scholar
Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.CrossRefGoogle Scholar