Multifractal Analysis, Liv\v{s}ic Rigidity, and Fluctuation Theorems for Axiom A Diffeomorphisms: The Pesin Formula and the Gallavotti-Cohen Symmetry

Abstract

This Part develops structural consequences of the thermodynamic formalism for Axiom A diffeomorphisms. The Pesin Entropy Formula equates the metric entropy of the SRB measure to the sum of positive Lyapunov exponents, with complete proofs of absolute continuity of conditional measures along unstable manifolds; the individual results are due to Sinai, Ruelle, Bowen, and Pesin. The Multifractal Formalism computes the Hausdorff dimension of Birkhoff average level sets via the Legendre transform of the pressure, extending earlier work of Barreira, Pesin, and Schmeling. The Liv\v{s}ic Theorem characterizes coboundaries through periodic orbit data with optimal H\"{o}lder regularity and an explicit norm bound in terms of the contraction rate and the H\"{o}lder exponent. The Gallavotti-Cohen Fluctuation Theorem establishes the linear symmetry relating the rate function at opposite values of the entropy production rate; for Axiom A diffeomorphisms the symmetry was established by Ruelle and by Maes, and we provide explicit bounds from the spectral gap. This Part constitutes Part VI, the final installment, of a six-part series on the thermodynamic formalism for hyperbolic dynamical systems.