# American Institue 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,, \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.
 [1] Luis Barreira. Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 279-305. doi: 10.3934/dcds.2006.16.279 [2] Yakov Pesin, Samuel Senti. Equilibrium measures for maps with inducing schemes. Journal of Modern Dynamics, 2008, 2 (3) : 397-430. doi: 10.3934/jmd.2008.2.397 [3] Vaughn Climenhaga. A note on two approaches to the thermodynamic formalism. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 995-1005. doi: 10.3934/dcds.2010.27.995 [4] Yakov Pesin. On the work of Sarig on countable Markov chains and thermodynamic formalism. Journal of Modern Dynamics, 2014, 8 (1) : 1-14. doi: 10.3934/jmd.2014.8.1 [5] Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1/2) : 131-164. doi: 10.3934/dcds.2008.22.131 [6] Yongluo Cao, De-Jun Feng, Wen Huang. The thermodynamic formalism for sub-additive potentials. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 639-657. doi: 10.3934/dcds.2008.20.639 [7] Anna Mummert. The thermodynamic formalism for almost-additive sequences. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 435-454. doi: 10.3934/dcds.2006.16.435 [8] Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Corrigendum to: Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593 [9] J. W. Neuberger. How to distinguish a local semigroup from a global semigroup. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11/12) : 5293-5303. doi: 10.3934/dcds.2013.33.5293 [10] Gerhard Keller. Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 313-334. doi: 10.3934/dcdss.2017015 [11] Eugen Mihailescu. Applications of thermodynamic formalism in complex dynamics on $\mathbb{P}^2$. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 821-836. doi: 10.3934/dcds.2001.7.821 [12] Omri M. Sarig. Bernoulli equilibrium states for surface diffeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 593-608. doi: 10.3934/jmd.2011.5.593 [13] Benoît Merlet, Morgan Pierre. Convergence to equilibrium for the backward Euler scheme and applications. Communications on Pure & Applied Analysis, 2010, 9 (3) : 685-702. doi: 10.3934/cpaa.2010.9.685 [14] 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 [15] Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285 [16] De-Jun Feng, Antti Käenmäki. Equilibrium states of the pressure function for products of matrices. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 699-708. doi: 10.3934/dcds.2011.30.699 [17] V. M. Gundlach, Yu. Kifer. Expansiveness, specification, and equilibrium states for random bundle transformations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 89-120. doi: 10.3934/dcds.2000.6.89 [18] Vaughn Climenhaga. Multifractal formalism derived from thermodynamics for general dynamical systems. Electronic Research Announcements, 2010, 17: 1-11. doi: 10.3934/era.2010.17.1 [19] Carles Bonet-Revés, Tere M-Seara. Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3545-3601. doi: 10.3934/dcds.2016.36.3545 [20] Pierre-Emmanuel Mazaré, Olli-Pekka Tossavainen, Daniel B. Work. Computing travel times from filtered traffic states. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 557-578. doi: 10.3934/dcdss.2014.7.557

2016 Impact Factor: 0.483