Linear response, and consequences for differentiability of statistical quantities and Multifractal Analysis

TL;DR

This paper establishes the differentiability of key statistical quantities in smooth expanding dynamical systems, providing formulas for their response to changes and implications for multifractal analysis.

Contribution

It proves the C^{r-1} differentiability of topological pressure, equilibrium states, and their densities, along with explicit linear and analytical response formulas.

Findings

Differentiability of statistical quantities with respect to dynamics and potential.

Explicit response formulas for topological pressure and equilibrium states.

Application to multifractal analysis and large deviations principles.

Abstract

In this article we initially fix ourselves to smooth (C^r) expanding dynamical systems. We prove the C^{r-1} differentiability of the topological pressure, equilibrium states and their densities with respect to smooth expanding dynamical systems and any smooth potential (C^{r-1}- linear response formula wiyh respect to the dynamics, and analytical response formula with respect to the potential). This is done by proving the regularity of the dominant eigenvalue of the transfer operator with respect to dynamics and potential. From that, we obtain strong consequences on the regularity of the dynamical system statistical properties, that apply in more general contexts. Indeed, we prove that the average and variance obtained from the central limit theorem vary $C^{r-1}$ with respect to the $C^{r}-$expanding dynamics and $C^{r}-$potential, and also, there is a large deviations principle with its rate $C^{r-1}$ with respect to the dynamics and potential. An application for multifractal analysis is given.