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.

[2]

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

[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.

[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.

[5]

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

[6]

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

[7]

G. Keller, Equilibrium States in Ergodic Theory, London Mathematical Society Student Texts, 42, Cambridge University Press, Cambridge, 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, Trans. Amer. Math. Soc., 352 (2000), 1889-1912. doi: 10.1090/S0002-9947-99-02479-4.

[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.

[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.

[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.

[13]

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

[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.

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.

[2]

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

[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.

[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.

[5]

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

[6]

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

[7]

G. Keller, Equilibrium States in Ergodic Theory, London Mathematical Society Student Texts, 42, Cambridge University Press, Cambridge, 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, Trans. Amer. Math. Soc., 352 (2000), 1889-1912. doi: 10.1090/S0002-9947-99-02479-4.

[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.

[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.

[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.

[13]

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

[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.

[1]

Luis Barreira. Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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]

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

[5]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2008, 20 (3) : 639-657. doi: 10.3934/dcds.2008.20.639

[7]

Anna Mummert. The thermodynamic formalism for almost-additive sequences. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593

[9]

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

[10]

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

[11]

Gerhard Keller. Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : 313-334. doi: 10.3934/dcdss.2017015

[12]

Eugen Mihailescu. Applications of thermodynamic formalism in complex dynamics on $\mathbb{P}^2$. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 821-836. doi: 10.3934/dcds.2001.7.821

[13]

J. W. Neuberger. How to distinguish a local semigroup from a global semigroup. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5293-5303. doi: 10.3934/dcds.2013.33.5293

[14]

Clark Butler, Kiho Park. Thermodynamic formalism of $ \text{GL}_2(\mathbb{R}) $-cocycles with canonical holonomies. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2141-2166. doi: 10.3934/dcds.2020356

[15]

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

[16]

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

[17]

Pedro Branco. A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions. Advances in Mathematics of Communications, 2021, 15 (1) : 113-130. doi: 10.3934/amc.2020046

[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]

V. M. Gundlach, Yu. Kifer. Expansiveness, specification, and equilibrium states for random bundle transformations. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 89-120. doi: 10.3934/dcds.2000.6.89

[20]

Alexander Arbieto, Luciano Prudente. Uniqueness of equilibrium states for some partially hyperbolic horseshoes. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 27-40. doi: 10.3934/dcds.2012.32.27

2020 Impact Factor: 0.929

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]