Quantitative statistical stability and convergence to equilibrium. An application to maps with indifferent fixed points

TL;DR

This paper establishes a link between the stability of fixed points in transfer operators and convergence to equilibrium, applying it to maps with indifferent fixed points to quantify how small deterministic changes affect their invariant measures.

Contribution

It introduces a general relation connecting fixed point stability and convergence to equilibrium, with explicit estimates for the dependence of invariant measures on perturbations for maps with indifferent fixed points.

Findings

Hölder continuity of invariant measures with respect to perturbations

Explicit estimates of Hölder exponents

Applicability to a broad class of maps with indifferent fixed points

Abstract

We show 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 large class of maps with indifferent fixed points. It turns out that the $L^1$ dependence of the a.c.i.m. on small suitable deterministic changes for these kind of maps is H\"older, with an exponent which is explicitly estimated.