American Institute of Mathematical Sciences

Advanced
  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,, \emph{Stoch. Dyn.}, 12 (2012).  doi: 10.1142/S0219493712500050.  Google Scholar

[2]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Lecture Notes in Mathematics, (1975).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

F. Hofbauer, Examples for the nonuniqueness of the equilibrium state,, \emph{Trans. Amer. Math. Soc.}, 228 (1977), 223.   Google Scholar

[7]

G. Keller, Equilibrium States in Ergodic Theory,, London Mathematical Society Student Texts, (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,, \emph{Trans. Amer. Math. Soc.}, 352 (2000), 1889.  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,, \emph{Nonlinearity}, 18 (2005), 2847.  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,, \emph{Ergodic Theory Dynam. Systems}, 31 (2011), 423.  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,, \emph{Ergodic Theory Dynam. Systems}, 29 (2009), 1917.  doi: 10.1017/S0143385708000941.  Google Scholar

[13]

K. Petersen, Ergodic Theory,, Corrected reprint of the 1983 original, (1983).   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Lecture Notes in Mathematics, (1975).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

F. Hofbauer, Examples for the nonuniqueness of the equilibrium state,, \emph{Trans. Amer. Math. Soc.}, 228 (1977), 223.   Google Scholar

[7]

G. Keller, Equilibrium States in Ergodic Theory,, London Mathematical Society Student Texts, (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,, \emph{Trans. Amer. Math. Soc.}, 352 (2000), 1889.  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,, \emph{Nonlinearity}, 18 (2005), 2847.  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,, \emph{Ergodic Theory Dynam. Systems}, 31 (2011), 423.  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,, \emph{Ergodic Theory Dynam. Systems}, 29 (2009), 1917.  doi: 10.1017/S0143385708000941.  Google Scholar

[13]

K. Petersen, Ergodic Theory,, Corrected reprint of the 1983 original, (1983).   Google Scholar

[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.  Google Scholar

[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]

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
[3]

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
[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]

Michael Jakobson, Lucia D. Simonelli. Countable Markov partitions suitable for thermodynamic formalism. Journal of Modern Dynamics, 2018, 13: 199-219. doi: 10.3934/jmd.2018018
[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]

L. Cioletti, E. Silva, M. Stadlbauer. Thermodynamic formalism for topological Markov chains on standard Borel spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6277-6298. doi: 10.3934/dcds.2019274
[13]

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
[14]

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
[15]

Dominic Veconi. Equilibrium states of almost Anosov diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 767-780. doi: 10.3934/dcds.2020061
[16]

Pedro Branco. A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions. Advances in Mathematics of Communications, 2019  doi: 10.3934/amc.2020046
[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]

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
[19]

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
[20]

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

2019 Impact Factor: 0.5

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences