American Institute of Mathematical Sciences

Advanced
August  2010, 27(3): 995-1005. doi: 10.3934/dcds.2010.27.995

A note on two approaches to the thermodynamic formalism

Vaughn Climenhaga 1,

1. 

Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802

Received  February 2009 Revised  January 2010 Published  March 2010

Inducing schemes provide a means of using symbolic dynamics to study equilibrium states of non-uniformly hyperbolic maps, but necessitate a solution to the liftability problem. One approach, due to Pesin and Senti, places conditions on the induced potential under which a unique equilibrium state exists among liftable measures, and then solves the liftability problem separately. Another approach, due to Bruin and Todd, places conditions on the original potential under which both problems may be solved simultaneously. These conditions include a bounded range condition, first introduced by Hofbauer and Keller. We compare these two sets of conditions and show that for many inducing schemes of interest, the conditions from the second approach are strictly stronger than the conditions from the first. We also show that the bounded range condition can be used to obtain Pesin and Senti's conditions for any inducing scheme with sufficiently slow growth of basic elements.
Keywords: Thermodynamic formalism, equilibrium states, inducing schemes, multimodal maps., unimodal maps.
Mathematics Subject Classification: Primary: 37D25, 37D35; Secondary: 37E05, 37E1.
Citation: 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
[1]

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
[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]

Peyman Eslami. Inducing schemes for multi-dimensional piecewise expanding maps. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 353-368. doi: 10.3934/dcds.2021120
[4]

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
[5]

Kaijen Cheng, Kenneth Palmer, Yuh-Jenn Wu. Period 3 and chaos for unimodal maps. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1933-1949. doi: 10.3934/dcds.2014.34.1933
[6]

Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617
[7]

Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685
[8]

Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo. Dynamics of continued fractions and kneading sequences of unimodal maps. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1313-1332. doi: 10.3934/dcds.2013.33.1313
[9]

Yan Gao. Monotonicity of entropy for unimodal real quadratic rational maps. Discrete and Continuous Dynamical Systems, 2022, 42 (11) : 5377-5386. doi: 10.3934/dcds.2022101
[10]

José M. Amigó, Ángel Giménez. Formulas for the topological entropy of multimodal maps based on min-max symbols. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3415-3434. doi: 10.3934/dcdsb.2015.20.3415
[11]

Viviane Baladi, Mark F. Demers. Thermodynamic formalism for dispersing billiards. Journal of Modern Dynamics, 2022, 18: 415-493. doi: 10.3934/jmd.2022013
[12]

Iryna Sushko, Anna Agliari, Laura Gardini. Bistability and border-collision bifurcations for a family of unimodal piecewise smooth maps. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 881-897. doi: 10.3934/dcdsb.2005.5.881
[13]

Chris Good, Robin Knight, Brian Raines. Countable inverse limits of postcritical $w$-limit sets of unimodal maps. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1059-1078. doi: 10.3934/dcds.2010.27.1059
[14]

Cesare Tronci. Momentum maps for mixed states in quantum and classical mechanics. Journal of Geometric Mechanics, 2019, 11 (4) : 639-656. doi: 10.3934/jgm.2019032
[15]

Lam Quoc Anh, Pham Thanh Duoc, Tran Ngoc Tam. Continuity of approximate solution maps to vector equilibrium problems. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1685-1699. doi: 10.3934/jimo.2017013
[16]

Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 35-61. doi: 10.3934/dcds.2008.22.35
[17]

Yi Yang, Robert J. Sacker. Periodic unimodal Allee maps, the semigroup property and the $\lambda$-Ricker map with Allee effect. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 589-606. doi: 10.3934/dcdsb.2014.19.589
[18]

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
[19]

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
[20]

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

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (110)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy