An optimal transportation approach to the decay of correlations for non-uniformly expanding maps

TL;DR

This paper introduces an optimal transportation-based method to analyze decay of correlations in non-uniformly expanding maps, establishing a Ruelle-Perron-Frobenius theorem under potential flatness without relying on traditional techniques.

Contribution

It presents a novel approach using optimal transportation and duality to prove decay of correlations and spectral gap results for non-uniformly expanding maps, avoiding Markov partitions and inducing.

Findings

Proves decay of transfer operators with flat potentials.

Shows density of spectral gap potentials on the circle.

Applies to Pomeau-Manneville maps and low-regularity expanding maps.

Abstract

We consider the transfer operators of non-uniformly expanding maps for potentials of various regularity, and show that a specific property of potentials ("flatness") implies a Ruelle-Perron-Frobenius Theorem and a decay of the transfer operator of the same speed than entailed by the constant potential. The method relies neither on Markov partitions nor on inducing, but on functional analysis and duality, through the simplest principles of optimal transportation. As an application, we notably show that for any map of the circle which is expanding outside an arbitrarily flat neutral point, the set of H{\"o}lder potentials exhibiting a spectral gap is dense in the uniform topology. The method applies in a variety of situation, including Pomeau-Manneville maps with regular enough potentials, or uniformly expanding maps of low regularity with their natural potential; we also recover in a united fashion variants of several previous results.