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

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

Renaud Leplaideur 1,

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

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, Stoch. Dyn., 12 (2012), 1250005, 34 pp. doi: 10.1142/S0219493712500050.  Google Scholar

[2]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin, 1975.  Google Scholar

[3]

H. Bruin and R. Leplaideur, Renormalization, thermodynamic formalism and quasi-crystals in subshifts, Comm. Math. Phys., 321 (2013), 209-247. doi: 10.1007/s00220-012-1651-4.  Google Scholar

[4]

J.-R. Chazottes and R. Leplaideur, Fluctuations of the $N$th return time for Axiom A diffeomorphisms, Discrete Contin. Dyn. Syst., 13 (2005), 399-411. doi: 10.3934/dcds.2005.13.399.  Google Scholar

[5]

Y. N. Dowker, Finite and $\sigma$-finite invariant measures, Ann. of Math. (2), 54 (1951), 595-608. doi: 10.2307/1969491.  Google Scholar

[6]

F. Hofbauer, Examples for the nonuniqueness of the equilibrium state, Trans. Amer. Math. Soc., 228 (1977), 223-241.  Google Scholar

[7]

G. Keller, Equilibrium States in Ergodic Theory, London Mathematical Society Student Texts, 42, Cambridge University Press, Cambridge, 1998.  Google Scholar

[8]

R. Leplaideur, Chaos: Butterflies also generate phase transitions and parallel universes, arXiv:1301.5413, 2013. Google Scholar

[9]

R. Leplaideur, Local product structure for equilibrium states, Trans. Amer. Math. Soc., 352 (2000), 1889-1912. doi: 10.1090/S0002-9947-99-02479-4.  Google Scholar

[10]

R. Leplaideur, A dynamical proof for the convergence of Gibbs measures at temperature zero, Nonlinearity, 18 (2005), 2847-2880. doi: 10.1088/0951-7715/18/6/023.  Google Scholar

[11]

R. Leplaideur, Thermodynamic formalism for a family of non-uniformly hyperbolic horseshoes and the unstable Jacobian, Ergodic Theory Dynam. Systems, 31 (2011), 423-447. doi: 10.1017/S0143385709001126.  Google Scholar

[12]

R. Leplaideur and I. Rios, On $t$-conformal measures and Hausdorff dimension for a family of non-uniformly hyperbolic horseshoes, Ergodic Theory Dynam. Systems, 29 (2009), 1917-1950. doi: 10.1017/S0143385708000941.  Google Scholar

[13]

K. Petersen, Ergodic Theory, Corrected reprint of the 1983 original, Cambridge Studies in Advanced Mathematics, 2, Cambridge University Press, Cambridge, 1989.  Google Scholar

[14]

Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys., 74 (1980), 189-197. doi: 10.1007/BF01197757.  Google Scholar

show all references

References:
[1]

A. Baraviera, R. Leplaideur and A. O. Lopes, The potential point of view for renormalization, Stoch. Dyn., 12 (2012), 1250005, 34 pp. doi: 10.1142/S0219493712500050.  Google Scholar

[2]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin, 1975.  Google Scholar

[3]

H. Bruin and R. Leplaideur, Renormalization, thermodynamic formalism and quasi-crystals in subshifts, Comm. Math. Phys., 321 (2013), 209-247. doi: 10.1007/s00220-012-1651-4.  Google Scholar

[4]

J.-R. Chazottes and R. Leplaideur, Fluctuations of the $N$th return time for Axiom A diffeomorphisms, Discrete Contin. Dyn. Syst., 13 (2005), 399-411. doi: 10.3934/dcds.2005.13.399.  Google Scholar

[5]

Y. N. Dowker, Finite and $\sigma$-finite invariant measures, Ann. of Math. (2), 54 (1951), 595-608. doi: 10.2307/1969491.  Google Scholar

[6]

F. Hofbauer, Examples for the nonuniqueness of the equilibrium state, Trans. Amer. Math. Soc., 228 (1977), 223-241.  Google Scholar

[7]

G. Keller, Equilibrium States in Ergodic Theory, London Mathematical Society Student Texts, 42, Cambridge University Press, Cambridge, 1998.  Google Scholar

[8]

R. Leplaideur, Chaos: Butterflies also generate phase transitions and parallel universes, arXiv:1301.5413, 2013. Google Scholar

[9]

R. Leplaideur, Local product structure for equilibrium states, Trans. Amer. Math. Soc., 352 (2000), 1889-1912. doi: 10.1090/S0002-9947-99-02479-4.  Google Scholar

[10]

R. Leplaideur, A dynamical proof for the convergence of Gibbs measures at temperature zero, Nonlinearity, 18 (2005), 2847-2880. doi: 10.1088/0951-7715/18/6/023.  Google Scholar

[11]

R. Leplaideur, Thermodynamic formalism for a family of non-uniformly hyperbolic horseshoes and the unstable Jacobian, Ergodic Theory Dynam. Systems, 31 (2011), 423-447. doi: 10.1017/S0143385709001126.  Google Scholar

[12]

R. Leplaideur and I. Rios, On $t$-conformal measures and Hausdorff dimension for a family of non-uniformly hyperbolic horseshoes, Ergodic Theory Dynam. Systems, 29 (2009), 1917-1950. doi: 10.1017/S0143385708000941.  Google Scholar

[13]

K. Petersen, Ergodic Theory, Corrected reprint of the 1983 original, Cambridge Studies in Advanced Mathematics, 2, Cambridge University Press, Cambridge, 1989.  Google Scholar

[14]

Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys., 74 (1980), 189-197. doi: 10.1007/BF01197757.  Google Scholar

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