Statistical Limit Theorems for Axiom A Diffeomorphisms: Exponential Mixing, Central Limit Theorem, and Large Deviations

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.