Differentiability of thermodynamical quantities in non-uniformly expanding dynamics

TL;DR

This paper investigates the differentiability and analytic properties of thermodynamical quantities in non-uniformly expanding dynamical systems without Markov assumptions, providing new insights into their stability and statistical behavior.

Contribution

It establishes differentiability and analyticity of pressure, equilibrium states, and Lyapunov exponents in non-uniformly expanding maps, with explicit formulas for derivatives.

Findings

Topological pressure is differentiable and analytic in the potential.

Equilibrium states and their entropy vary continuously with dynamics.

Correlation functions are differentiable and converge to zero in the studied systems.

Abstract

In this paper we study the ergodic theory of a robust non-uniformly expanding maps where no Markov assumption is required. We prove that the topological pressure is differentiable as a function of the dynamics and analytic with respect to the potential. Moreover we not only prove the continuity of the equilibrium states and their metric entropy as well as the differentiability of the maximal entropy measure and extremal Lyapunov exponents with respect to the dynamics. We also prove a local large deviations principle and central limit theorem and show that the rate function, mean and variance vary continuously with respect to observables, potentials and dynamics. Finally, we show that the correlation function associated to the maximal entropy measure is differentiable with respect to the dynamics and it is $C^1$-convergent to zero. In addition, precise formulas for the derivatives of thermodynamical quantities are given.