June  2006, 16(2): 435-454. doi: 10.3934/dcds.2006.16.435

The thermodynamic formalism for almost-additive sequences

1. 

Penn State University, McAllister Building, State College, PA 16802, United States

Received  July 2005 Revised  December 2005 Published  July 2006

We study the nonadditive thermodynamic formalism for the class of almost-additive sequences of potentials. We define the topological pressure $P_Z(\Phi)$ of an almost-additive sequence $\Phi$, on a set $Z$. We give conditions which allow us to establish a variational principle for the topological pressure. We state conditions for the existence and uniqueness of equilibrium measures, and for subshifts of finite type the existence and uniqueness of Gibbs measures. Finally, we compare the results for almost-additive sequences to the thermodynamic formalism for the classical (additive) case [10] [11] [3], the sequences studied by Barreira [1], Falconer [5], and that of Feng and Lau [7], [6].
Citation: 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
[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]

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

[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]

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

[8]

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

[9]

Zhihui Yuan. Multifractal analysis of random weak Gibbs measures. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5367-5405. doi: 10.3934/dcds.2017234

[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]

Guizhen Cui, Yunping Jiang, Anthony Quas. Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 535-552. doi: 10.3934/dcds.1999.5.535

[14]

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

[15]

Barry Simon. Equilibrium measures and capacities in spectral theory. Inverse Problems & Imaging, 2007, 1 (4) : 713-772. doi: 10.3934/ipi.2007.1.713

[16]

Yair Daon. Bernoullicity of equilibrium measures on countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4003-4015. doi: 10.3934/dcds.2013.33.4003

[17]

Eugen Mihailescu. Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2485-2502. doi: 10.3934/dcds.2012.32.2485

[18]

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

[19]

Vítor Araújo. Semicontinuity of entropy, existence of equilibrium states and continuity of physical measures. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 371-386. doi: 10.3934/dcds.2007.17.371

[20]

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

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (23)

Other articles
by authors

[Back to Top]