2012, 32(1): 27-40. doi: 10.3934/dcds.2012.32.27

Uniqueness of equilibrium states for some partially hyperbolic horseshoes

1. 

Instituto de Matemática - UFRJ, Av. Athos da Silveira Ramos 149, Cidade Universitária - Ilha do Fundão, P.O. Box 68530. Rio de Janeiro - RJ, Brazil, Brazil

Received  July 2010 Revised  May 2011 Published  September 2011

In this note, we consider a partially hyperbolic horseshoe and prove uniqueness of equilibrium states for a class of potentials. In particular we obtain that there exists a unique maximal entropy measure.
Citation: Alexander Arbieto, Luciano Prudente. Uniqueness of equilibrium states for some partially hyperbolic horseshoes. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 27-40. doi: 10.3934/dcds.2012.32.27
References:
[1]

J. Alves and V. Araújo, Random perturbations of non-uniformly expanding maps,, Astérisque, 286 (2003), 25.

[2]

J. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding,, Invent. Math., 140 (2000), 351. doi: 10.1007/s002220000057.

[3]

A. Arbieto, C. Matheus and K. Oliveira, Equilibrium states for random non-uniformly expanding maps,, Nonlinearity, 17 (2004), 581. doi: 10.1088/0951-7715/17/2/013.

[4]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting,, Israel J. of Math., 115 (2000), 157. doi: 10.1007/BF02810585.

[5]

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms,, Trans. Amer. Math. Soc., 154 (1971), 377.

[6]

R. Bowen, Some systems with unique equilibrium states,, Math. Systems Theory, 8 (): 1974.

[7]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401. doi: 10.1090/S0002-9947-1971-0274707-X.

[8]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphism,", Springer Lecture Notes in Math., 470 (1975).

[9]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows,, Invent. Math., 29 (1975), 181. doi: 10.1007/BF01389848.

[10]

H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics,, Comm. Math. Phys., 168 (1995), 571. doi: 10.1007/BF02101844.

[11]

H. Bruin and G. Keller, Equilibrium states for S-unimodal maps,, Ergodic Theory Dynam. Systems, 18 (1998), 765. doi: 10.1017/S0143385798108337.

[12]

H. Bruin and M. Todd, Equilibrium states for interval maps: The potential $-t log\|Df\|$,, Ann. Sci. École Norm. Sup., 42 (2009), 559.

[13]

J. Buzzi, Thermodynamical formalism for piecewise invertible maps: Absolutely continuous invariant measures as equilibrium states,, Proc. Sympos. Pure Math., 69 (2001), 749.

[14]

J. Buzzi, T. Fisher, M. Sambarino and C. Vásquez, Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems,, Ergodic Theory and Dynamical Systems, ().

[15]

J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps,, Ergodic Theory Dynam. Systems, 23 (2003), 1383. doi: 10.1017/S0143385703000087.

[16]

L. J. Díaz and T. Fisher, Symbolic extensions for partially hyperbolic diffeomorphisms,, Discrete and Continuous Dynamical Systems, 29 (2011), 1419.

[17]

L. J. Díaz, V. Horita, M. Sambarino and I. Rios, Destroying horseshoes via heterodimensional cycles: Generating bifurcations inside homoclinic classes,, Ergodic Theory and Dynamical Systems, 29 (2009), 433. doi: 10.1017/S0143385708080346.

[18]

Haydn N.T.A. and D. Ruelle, Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification,, Commun. Math. Phys., 148 (1992), 155. doi: 10.1007/BF02102369.

[19]

F. Hofbauer, The topological entropy of a transformation $x\mapsto ax(1-x)$,, Monatsh. Math., 90 (1980), 117. doi: 10.1007/BF01303262.

[20]

G. Iommi and M. Todd, Natural equilibrium states for multimodal maps,, Commun. Math. Phys., 300 (2010), 65. doi: 10.1007/s00220-010-1112-x.

[21]

R. Israel, "Convexity in the Theory of Lattice Gases,", Princeton University Press, (1979).

[22]

G. Keller, Lifting measures to Markov extensions,, Monatsh. Math., 108 (1989), 183.

[23]

F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations,, J. London Math. Soc., 16 (1977), 568. doi: 10.1112/jlms/s2-16.3.568.

[24]

R. Leplaideur, K. Oliveira and I. Rios, Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes,, Nonlinearity, 19 (2006), 2667. doi: 10.1088/0951-7715/19/11/009.

[25]

S. E. Newhouse, Continuity properties of entropy,, Annals of Mathematics, 129 (1989), 215. doi: 10.2307/1971492.

[26]

K. Oliveira, Equilibrium states for non-uniformly expanding maps,, Ergodic Theory & Dynamical Systems, 23 (2003), 1891. doi: 10.1017/S0143385703000257.

[27]

Y. Pesin and S. Senti, Equilibrium measures for maps with inducing schemes,, Journal of Modern Dynamics, 2 (2008), 397.

[28]

V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure,, Russian Math. Surveys, 22 (1967), 3.

[29]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics,", Encyclopedia of Mathematics and its Applications, 5 (1978).

[30]

P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps,, Annales de l Institut Henri Poincaré. Analyse non Linéaire, 27 (2010), 555.

[31]

W. Cowieson and L. S. Young, SRB measures as zero-noise limits,, Ergodic Theory Dynamic Systems, 25 (2005), 1115. doi: 10.1017/S0143385704000604.

show all references

References:
[1]

J. Alves and V. Araújo, Random perturbations of non-uniformly expanding maps,, Astérisque, 286 (2003), 25.

[2]

J. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding,, Invent. Math., 140 (2000), 351. doi: 10.1007/s002220000057.

[3]

A. Arbieto, C. Matheus and K. Oliveira, Equilibrium states for random non-uniformly expanding maps,, Nonlinearity, 17 (2004), 581. doi: 10.1088/0951-7715/17/2/013.

[4]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting,, Israel J. of Math., 115 (2000), 157. doi: 10.1007/BF02810585.

[5]

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms,, Trans. Amer. Math. Soc., 154 (1971), 377.

[6]

R. Bowen, Some systems with unique equilibrium states,, Math. Systems Theory, 8 (): 1974.

[7]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401. doi: 10.1090/S0002-9947-1971-0274707-X.

[8]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphism,", Springer Lecture Notes in Math., 470 (1975).

[9]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows,, Invent. Math., 29 (1975), 181. doi: 10.1007/BF01389848.

[10]

H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics,, Comm. Math. Phys., 168 (1995), 571. doi: 10.1007/BF02101844.

[11]

H. Bruin and G. Keller, Equilibrium states for S-unimodal maps,, Ergodic Theory Dynam. Systems, 18 (1998), 765. doi: 10.1017/S0143385798108337.

[12]

H. Bruin and M. Todd, Equilibrium states for interval maps: The potential $-t log\|Df\|$,, Ann. Sci. École Norm. Sup., 42 (2009), 559.

[13]

J. Buzzi, Thermodynamical formalism for piecewise invertible maps: Absolutely continuous invariant measures as equilibrium states,, Proc. Sympos. Pure Math., 69 (2001), 749.

[14]

J. Buzzi, T. Fisher, M. Sambarino and C. Vásquez, Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems,, Ergodic Theory and Dynamical Systems, ().

[15]

J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps,, Ergodic Theory Dynam. Systems, 23 (2003), 1383. doi: 10.1017/S0143385703000087.

[16]

L. J. Díaz and T. Fisher, Symbolic extensions for partially hyperbolic diffeomorphisms,, Discrete and Continuous Dynamical Systems, 29 (2011), 1419.

[17]

L. J. Díaz, V. Horita, M. Sambarino and I. Rios, Destroying horseshoes via heterodimensional cycles: Generating bifurcations inside homoclinic classes,, Ergodic Theory and Dynamical Systems, 29 (2009), 433. doi: 10.1017/S0143385708080346.

[18]

Haydn N.T.A. and D. Ruelle, Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification,, Commun. Math. Phys., 148 (1992), 155. doi: 10.1007/BF02102369.

[19]

F. Hofbauer, The topological entropy of a transformation $x\mapsto ax(1-x)$,, Monatsh. Math., 90 (1980), 117. doi: 10.1007/BF01303262.

[20]

G. Iommi and M. Todd, Natural equilibrium states for multimodal maps,, Commun. Math. Phys., 300 (2010), 65. doi: 10.1007/s00220-010-1112-x.

[21]

R. Israel, "Convexity in the Theory of Lattice Gases,", Princeton University Press, (1979).

[22]

G. Keller, Lifting measures to Markov extensions,, Monatsh. Math., 108 (1989), 183.

[23]

F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations,, J. London Math. Soc., 16 (1977), 568. doi: 10.1112/jlms/s2-16.3.568.

[24]

R. Leplaideur, K. Oliveira and I. Rios, Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes,, Nonlinearity, 19 (2006), 2667. doi: 10.1088/0951-7715/19/11/009.

[25]

S. E. Newhouse, Continuity properties of entropy,, Annals of Mathematics, 129 (1989), 215. doi: 10.2307/1971492.

[26]

K. Oliveira, Equilibrium states for non-uniformly expanding maps,, Ergodic Theory & Dynamical Systems, 23 (2003), 1891. doi: 10.1017/S0143385703000257.

[27]

Y. Pesin and S. Senti, Equilibrium measures for maps with inducing schemes,, Journal of Modern Dynamics, 2 (2008), 397.

[28]

V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure,, Russian Math. Surveys, 22 (1967), 3.

[29]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics,", Encyclopedia of Mathematics and its Applications, 5 (1978).

[30]

P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps,, Annales de l Institut Henri Poincaré. Analyse non Linéaire, 27 (2010), 555.

[31]

W. Cowieson and L. S. Young, SRB measures as zero-noise limits,, Ergodic Theory Dynamic Systems, 25 (2005), 1115. doi: 10.1017/S0143385704000604.

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