SRB Entropy of Markov Transformations

In the family of piecewise differentiable expanding Markov transformations, every map preserves a unique absolutely continuous invariant measure. The measure theoretical entropy with respect to this invariant measure depends on the map differentiably. We prove that this entropy functional does not h...

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Bibliographic Details
Published inJournal of statistical physics Vol. 188; no. 3
Main Authors Jiang, Miaohua, Lopez, Marco
Format Journal Article
LanguageEnglish
Published New York Springer US 01.09.2022
Springer
Springer Nature B.V
Subjects
Critical point
Invariants
Mathematical and Computational Physics
Physical Chemistry
Physics
Physics and Astronomy
Quantum Physics
Statistical Physics and Dynamical Systems
Theoretical
Thermodynamics
Toruses
Transformations (mathematics)
Secondary: 37A35
82B05
Markov transformations
Primary: 37E05
Entropy
37A60
80-10
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Summary:In the family of piecewise differentiable expanding Markov transformations, every map preserves a unique absolutely continuous invariant measure. The measure theoretical entropy with respect to this invariant measure depends on the map differentiably. We prove that this entropy functional does not have any nontrivial local critical points. The only critical points are where the functional reaches its global maximum. We also observe that the same statement holds true for the family of sufficiently smooth volume-preserving Anosov maps on a torus. Based on Gallavotti–Cohen Chaotic Hypothesis, these families of chaotic maps may serve as mathematical models of thermodynamic systems where the second law of thermodynamics can be rigorously proved.
ISSN:0022-4715
1572-9613
DOI:10.1007/s10955-022-02954-y