# American Institute of Mathematical Sciences

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

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.
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

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##### 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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