Five characterizations of Gibbs measures for Hölder potentials on topologically mixing subshifts of finite type are equivalent in a single theorem with explicit constants, plus derived spectral gaps, stability, and limit theorems.
The Convex-Analytic Structure of Thermodynamic Equilibrium: Pressure, Subdifferentials, and Phase Transitions
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abstract
We develop the convex-analytic structure of the thermodynamic formalism for continuous maps on compact metric spaces. The pressure functional is the Legendre-Fenchel transform of the negative entropy, and the biconjugate recovery of the entropy from the pressure establishes a complete duality. Equilibrium states are elements of the subdifferential of the pressure, uniqueness of equilibrium states corresponds to G\^{a}teaux differentiability, and first-order phase transitions correspond to non-differentiability. For systems with specification and H\"{o}lder potentials, the pressure is Fr\'{e}chet differentiable in the H\"{o}lder norm, and the second derivative of the pressure equals the asymptotic variance of the Birkhoff sums. We prove a universal variational principle that unifies the classical additive, the subadditive, and the relative variational principles through a single theorem on convex functionals satisfying convexity, lower semi-continuity, coercivity, and cocycle invariance. Extensions to systems with the specification property and to non-compact spaces under coercivity conditions are included, with applications to countable Markov shifts via Sarig's recurrence classification. This Part constitutes Part II 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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Gibbs Measures on Subshifts of Finite Type: Five Equivalent Characterizations with Explicit Constants
Five characterizations of Gibbs measures for Hölder potentials on topologically mixing subshifts of finite type are equivalent in a single theorem with explicit constants, plus derived spectral gaps, stability, and limit theorems.
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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