American Institute of Mathematical Sciences

Advanced
June  2006, 16(2): 279-305. doi: 10.3934/dcds.2006.16.279

Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures

Luis Barreira 1,

1. 

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa

Received  December 2005 Published  July 2006

The nonadditive thermodynamic formalism is a generalization of the classical thermodynamic formalism, in which the topological pressure of a single function $\phi$ is replaced by the topological pressure of a sequence of functions $\Phi=(\phi_n)_n$. The theory also includes a variational principle for the topological pressure, although with restrictive assumptions on $\Phi$. Our main objective is to provide a new class of sequences, the so-called almost additive sequences, for which it is possible not only to establish a variational principle, but also to discuss the existence and uniqueness of equilibrium and Gibbs measures. In addition, we give several characterizations of the invariant Gibbs measures, also in terms of an averaging procedure over the periodic points.
Keywords: thermodynamic formalism., Almost additive sequences.
Mathematics Subject Classification: Primary: 37D3.
Citation: 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
[1]

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
[2]

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
[3]

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
[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]

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
[7]

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
[8]

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
[9]

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
[10]

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
[11]

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
[12]

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
[13]

Julien Barral, Yan-Hui Qu. Localized asymptotic behavior for almost additive potentials. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 717-751. doi: 10.3934/dcds.2012.32.717
[14]

Büşra Özden, Oǧuz Yayla. Partial direct product difference sets and almost quaternary sequences. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021010
[15]

Yang Yang, Xiaohu Tang, Guang Gong. New almost perfect, odd perfect, and perfect sequences from difference balanced functions with d-form property. Advances in Mathematics of Communications, 2017, 11 (1) : 67-76. doi: 10.3934/amc.2017002
[16]

Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185
[17]

Benedict Leimkuhler, Charles Matthews, Tiffany Vlaar. Partitioned integrators for thermodynamic parameterization of neural networks. Foundations of Data Science, 2019, 1 (4) : 457-489. doi: 10.3934/fods.2019019
[18]

Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004
[19]

Jordan Emme. Hermodynamic formalism and k-bonacci substitutions. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3701-3719. doi: 10.3934/dcds.2017157
[20]

Katarzyna Grabowska. Lagrangian and Hamiltonian formalism in Field Theory: A simple model. Journal of Geometric Mechanics, 2010, 2 (4) : 375-395. doi: 10.3934/jgm.2010.2.375

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (157)
  • HTML views (0)
  • Cited by (37)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy