Thermodynamic formalism for discontinuous maps and statistical properties of their equilibrium states

Abstract

In this article, we develop a functional-analytic framework to establish existence, uniqueness, regularity of disintegration, and statistical properties of equilibrium states for a broad class of dynamical systems, potentially discontinuous and not necessarily locally invertible. Our approach is applied to a family of piecewise partially hyperbolic maps and associated classes of potentials. We further prove several statistical limit theorems, including exponential decay of correlations, and propose related questions and conjectures. A collection of examples illustrating the applicability of our results is provided, including partially hyperbolic attractors over horseshoes, discontinuous systems, non-invertible dynamical systems admitting a semi-conjugacy to intermittent maps such as the Manneville--Pomeau map, and fat solenoidal attractors.