Mixing speed and stability of SRB measures through optimal transportation

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).