Quantitative statistical stability and speed of convergence to equilibrium for partially hyperbolic skew products

TL;DR

This paper investigates the stability and convergence to equilibrium of partially hyperbolic skew products, establishing quantitative estimates on how physical measures depend on small deterministic perturbations, with applications to toral extensions.

Contribution

It introduces a relation between fixed point stability of transfer operators and convergence to equilibrium, applying it to partially hyperbolic skew products and power law mixing toral extensions.

Findings

Physical measures depend Holder continuously on perturbations

Explicit estimates of stability exponents based on system arithmetics

Examples of non-differentiable dependence of measures on perturbations

Abstract

We consider a general relation between fixed point stability of suitably perturbed transfer operators and convergence to equilibrium (a notion which is strictly related to decay of correlations). We apply this relation to deterministic perturbations of a class of (piecewise) partially hyperbolic skew products whose behavior on the preserved fibration is dominated by the expansion of the base map. In particular we apply the results to power law mixing toral extensions. It turns out that in this case, the dependence of the physical measure on small deterministic perturbations, in a suitable anisotropic metric is at least Holder continuous, with an exponent which is explicitly estimated depending on the arithmetical properties of the system. We show explicit examples of toral extensions having actually Holder stability and non differentiable dependence of the physical measure on perturbations.