Thermodynamic pressure is the Legendre-Fenchel transform of negative entropy; equilibrium states are its subdifferentials, phase transitions mark non-differentiability, and a universal variational principle unifies additive, subadditive, and relative cases.
Gibbs Measures on Subshifts of Finite Type: Five Equivalent Characterizations with Explicit Constants
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abstract
We prove that five characterizations of Gibbs measures for H\"{o}lder potentials on topologically mixing subshifts of finite type are equivalent: the Jacobian condition, the classical cylinder-based Gibbs property, the eigenmeasure of the Ruelle transfer operator, the variational equilibrium state, and the minimizer of the large deviations rate function. The equivalence is established in a single theorem with explicit constants expressed in terms of the H\"{o}lder exponent, the potential norm, the alphabet size, and the mixing time. The proof yields explicit spectral gap estimates for the transfer operator via the Birkhoff cone contraction technique, Lipschitz stability of the Gibbs measure in Wasserstein distance under perturbation of the potential, and statistical limit theorems including a central limit theorem with Berry-Esseen bounds and a large deviations principle. This Part constitutes Part I of a six-part series on the thermodynamic formalism for hyperbolic dynamical systems.
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math.DS 5years
2026 5verdicts
UNVERDICTED 5representative citing papers
Derives Volume Lemma, exponential correlations decay, CLT with optimal Berry-Esseen bounds, ASIP, and LDP for Axiom A equilibrium states from a single spectral gap with explicit hyperbolicity dependence.
Proves structural stability, quasi-compactness of transfer operators with spectral gap, SRB measures as unique equilibrium states for the geometric potential, and Pesin entropy formula for Axiom A diffeomorphisms, yielding the Gibbs Equivalence Theorem via Markov partition coding.
The paper establishes quantitative versions of the stable manifold theorem, shadowing lemma, spectral decomposition, Markov partitions, and Hölder coding for Axiom A diffeomorphisms to transfer symbolic dynamics to the smooth setting.
For Axiom A diffeomorphisms the metric entropy of the SRB measure equals the sum of positive Lyapunov exponents, Birkhoff level sets have Hausdorff dimension given by the Legendre transform of pressure, coboundaries are characterized by periodic data, and the entropy production rate function obeys a
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The Convex-Analytic Structure of Thermodynamic Equilibrium: Pressure, Subdifferentials, and Phase Transitions
Thermodynamic pressure is the Legendre-Fenchel transform of negative entropy; equilibrium states are its subdifferentials, phase transitions mark non-differentiability, and a universal variational principle unifies additive, subadditive, and relative cases.
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Statistical Limit Theorems for Axiom A Diffeomorphisms: Exponential Mixing, Central Limit Theorem, and Large Deviations
Derives Volume Lemma, exponential correlations decay, CLT with optimal Berry-Esseen bounds, ASIP, and LDP for Axiom A equilibrium states from a single spectral gap with explicit hyperbolicity dependence.
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Transfer Operators and SRB Measures for Axiom A Diffeomorphisms: Spectral Gap, Structural Stability, and the Gibbs Equivalence Theorem
Proves structural stability, quasi-compactness of transfer operators with spectral gap, SRB measures as unique equilibrium states for the geometric potential, and Pesin entropy formula for Axiom A diffeomorphisms, yielding the Gibbs Equivalence Theorem via Markov partition coding.
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Uniform Hyperbolicity and Symbolic Dynamics: Markov Partitions, Shadowing, and the Coding of Axiom A Diffeomorphisms
The paper establishes quantitative versions of the stable manifold theorem, shadowing lemma, spectral decomposition, Markov partitions, and Hölder coding for Axiom A diffeomorphisms to transfer symbolic dynamics to the smooth setting.
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Multifractal Analysis, Liv\v{s}ic Rigidity, and Fluctuation Theorems for Axiom A Diffeomorphisms: The Pesin Formula and the Gallavotti-Cohen Symmetry
For Axiom A diffeomorphisms the metric entropy of the SRB measure equals the sum of positive Lyapunov exponents, Birkhoff level sets have Hausdorff dimension given by the Legendre transform of pressure, coboundaries are characterized by periodic data, and the entropy production rate function obeys a