Optimal transport and dynamics of expanding circle maps acting on measures

TL;DR

This paper explores the dynamics of expanding circle maps on measures using optimal transport, revealing infinite-dimensional deformations of invariant measures and providing counterexamples to a conjecture, while also establishing positive metric mean dimension.

Contribution

It introduces a novel analysis of measure dynamics via optimal transport, computing derivatives at invariant measures and demonstrating infinite multiplicity eigenvalues.

Findings

Infinite multiplicity eigenvalue at 1 for the derivative

Existence of many deformations into nearly invariant measures

Positive metric mean dimension for the action

Abstract

Using optimal transport we study some dynamical properties of expanding circle maps acting on measures by push-forward. Using the definition of the tangent space to the space of measures introduced by Gigli, their derivative at the unique absolutely continuous invariant measure is computed. In particular it is shown that 1 is an eigenvalue of infinite multiplicity, so that the invariant measure admits many deformations into nearly invariant ones. As a consequence, we obtain counter-examples to an infinitesimal version of Furstenberg's conjecture. We also prove that this action has positive metric mean dimension with respect to the Wasserstein metric.