From local to global equilibrium states: Thermodynamic formalism via an inducing scheme

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January 2014, 21: 72-79. doi: 10.3934/era.2014.21.72

From local to global equilibrium states: Thermodynamic formalism via an inducing scheme

1.	Laboratoire de Mathématiques de Bretagne Atlantique, UMR 6205, Université de Brest, 6, avenue Victor Le Gorgeu, C.S. 93837, France

Received October 2013 Revised December 2013 Published May 2014

Abstract
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We present a method to construct equilibrium states via inducing. This method can be used for some non-uniformly hyperbolic dynamical systems and for non-Hölder continuous potentials. It allows us to prove the existence of phase transition.

Mathematics Subject Classification: 37D3.

Citation: Renaud Leplaideur. From local to global equilibrium states: Thermodynamic formalism via an inducing scheme. Electronic Research Announcements, 2014, 21: 72-79. doi: 10.3934/era.2014.21.72

References:

[1]	A. Baraviera, R. Leplaideur and A. O. Lopes, The potential point of view for renormalization,, \emph{Stoch. Dyn.}, 12 (2012). doi: 10.1142/S0219493712500050.
[2]	R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Lecture Notes in Mathematics, (1975).
[3]	H. Bruin and R. Leplaideur, Renormalization, thermodynamic formalism and quasi-crystals in subshifts,, \emph{Comm. Math. Phys.}, 321 (2013), 209. doi: 10.1007/s00220-012-1651-4.
[4]	J.-R. Chazottes and R. Leplaideur, Fluctuations of the $N$th return time for Axiom A diffeomorphisms,, \emph{Discrete Contin. Dyn. Syst.}, 13 (2005), 399. doi: 10.3934/dcds.2005.13.399.
[5]	Y. N. Dowker, Finite and $\sigma$-finite invariant measures,, \emph{Ann. of Math. (2)}, 54 (1951), 595. doi: 10.2307/1969491.
[6]	F. Hofbauer, Examples for the nonuniqueness of the equilibrium state,, \emph{Trans. Amer. Math. Soc.}, 228 (1977), 223.
[7]	G. Keller, Equilibrium States in Ergodic Theory,, London Mathematical Society Student Texts, (1998).
[8]	R. Leplaideur, Chaos: Butterflies also generate phase transitions and parallel universes,, \arXiv{1301.5413}, (2013).
[9]	R. Leplaideur, Local product structure for equilibrium states,, \emph{Trans. Amer. Math. Soc.}, 352 (2000), 1889. doi: 10.1090/S0002-9947-99-02479-4.
[10]	R. Leplaideur, A dynamical proof for the convergence of Gibbs measures at temperature zero,, \emph{Nonlinearity}, 18 (2005), 2847. doi: 10.1088/0951-7715/18/6/023.
[11]	R. Leplaideur, Thermodynamic formalism for a family of non-uniformly hyperbolic horseshoes and the unstable Jacobian,, \emph{Ergodic Theory Dynam. Systems}, 31 (2011), 423. doi: 10.1017/S0143385709001126.
[12]	R. Leplaideur and I. Rios, On $t$-conformal measures and Hausdorff dimension for a family of non-uniformly hyperbolic horseshoes,, \emph{Ergodic Theory Dynam. Systems}, 29 (2009), 1917. doi: 10.1017/S0143385708000941.
[13]	K. Petersen, Ergodic Theory,, Corrected reprint of the 1983 original, (1983).
[14]	Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems,, \emph{Comm. Math. Phys.}, 74 (1980), 189. doi: 10.1007/BF01197757.

show all references

References:

[1]	A. Baraviera, R. Leplaideur and A. O. Lopes, The potential point of view for renormalization,, \emph{Stoch. Dyn.}, 12 (2012). doi: 10.1142/S0219493712500050.
[2]	R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Lecture Notes in Mathematics, (1975).
[3]	H. Bruin and R. Leplaideur, Renormalization, thermodynamic formalism and quasi-crystals in subshifts,, \emph{Comm. Math. Phys.}, 321 (2013), 209. doi: 10.1007/s00220-012-1651-4.
[4]	J.-R. Chazottes and R. Leplaideur, Fluctuations of the $N$th return time for Axiom A diffeomorphisms,, \emph{Discrete Contin. Dyn. Syst.}, 13 (2005), 399. doi: 10.3934/dcds.2005.13.399.
[5]	Y. N. Dowker, Finite and $\sigma$-finite invariant measures,, \emph{Ann. of Math. (2)}, 54 (1951), 595. doi: 10.2307/1969491.
[6]	F. Hofbauer, Examples for the nonuniqueness of the equilibrium state,, \emph{Trans. Amer. Math. Soc.}, 228 (1977), 223.
[7]	G. Keller, Equilibrium States in Ergodic Theory,, London Mathematical Society Student Texts, (1998).
[8]	R. Leplaideur, Chaos: Butterflies also generate phase transitions and parallel universes,, \arXiv{1301.5413}, (2013).
[9]	R. Leplaideur, Local product structure for equilibrium states,, \emph{Trans. Amer. Math. Soc.}, 352 (2000), 1889. doi: 10.1090/S0002-9947-99-02479-4.
[10]	R. Leplaideur, A dynamical proof for the convergence of Gibbs measures at temperature zero,, \emph{Nonlinearity}, 18 (2005), 2847. doi: 10.1088/0951-7715/18/6/023.
[11]	R. Leplaideur, Thermodynamic formalism for a family of non-uniformly hyperbolic horseshoes and the unstable Jacobian,, \emph{Ergodic Theory Dynam. Systems}, 31 (2011), 423. doi: 10.1017/S0143385709001126.
[12]	R. Leplaideur and I. Rios, On $t$-conformal measures and Hausdorff dimension for a family of non-uniformly hyperbolic horseshoes,, \emph{Ergodic Theory Dynam. Systems}, 29 (2009), 1917. doi: 10.1017/S0143385708000941.
[13]	K. Petersen, Ergodic Theory,, Corrected reprint of the 1983 original, (1983).
[14]	Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems,, \emph{Comm. Math. Phys.}, 74 (1980), 189. doi: 10.1007/BF01197757.

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