Mixing speed and stability of SRB measures through optimal transportation

TL;DR

This paper introduces a novel optimal transport approach to analyze the mixing speed and stability of SRB measures in Anosov diffeomorphisms, demonstrating exponential decay of correlations and quantitative stability results.

Contribution

It develops a new metric based on optimal transport along stable foliations, providing fresh insights into SRB measure stability and correlation decay in hyperbolic systems.

Findings

SRB measures exhibit exponential decay of correlations for stable and unstable H{"o}lder observables.

The proposed metric makes the diffeomorphism act as a contraction, leading to stability results.

The map from diffeomorphisms to SRB measures is locally H{"o}lder-continuous.

Abstract

It is well-known that the SRB measure of a $C^{1+\alpha}$ Anosov diffeomorphism has exponential decay of correlations with respect to H{\"o}lder-continuous observables. We propose a new approach to this phenomenon, based on optimal transport. More precisely, we define a space of measures having absolutely continuous disintegrations with respect to some foliation close to the unstable foliation of the map, endowed with a variant of the Wasserstein metric where mass is only allowed to be transported along the diffeomorphism's stable foliation. We show that this metric is indeed finite on that space, and use that the construction makes the diffeomorphism act as a contraction to deduce two corollaries. First, the SRB measure has exponential decay of correlation with respect to pairs of observable that are only asked to be H{\"o}lder-continuous \emph{in the stable, respectively unstable direction}, but can be discontinuous overall. Then, we prove quantitative statistical stability: the map sending a $C^{1+\alpha}$ Anosov diffeomorphism to its SRB measure is locally H{\"o}lder-continuous (using the $C^1$ metric for diffeomorphisms and the usual Wasserstein metric for measures).