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.
Statistical Limit Theorems for Axiom A Diffeomorphisms: Exponential Mixing, Central Limit Theorem, and Large Deviations
5 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We establish statistical limit theorems for equilibrium states of Axiom A diffeomorphisms, derived from the spectral gap of the Ruelle transfer operator established in Part I (Thiam2026a) and transferred to smooth dynamics through the Markov partition coding of Part III (Thiam2026c). This Part contains five Main Theorems. The first proves the Volume Lemma with explicit two-sided bounds on the Riemannian volume of dynamical Bowen balls in terms of Birkhoff sums of the geometric potential. The second establishes exponential decay of correlations with explicit mixing rates computed from the spectral gap of the normalized transfer operator. The third proves the Central Limit Theorem with Berry-Esseen bounds at the optimal rate, with an explicit spectral formula for the asymptotic variance and a characterization of its degeneracy through the Liv\v{s}ic coboundary condition. The fourth establishes the Almost Sure Invariance Principle, providing pathwise Brownian approximation with polynomial error via the martingale embedding method. The fifth proves a large deviations principle with rate function given by the Legendre transform of the pressure. The individual results are due to Sinai, Ruelle, Ratner, Denker-Philipp, Gou\"{e}zel, Kifer, Melbourne-Nicol, and Young; the contribution is their derivation from a single spectral mechanism with explicit dependence on hyperbolicity data. This Part constitutes Part V 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
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.
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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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
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