Geodesic stretch, pressure metric and marked length spectrum rigidity

TL;DR

This paper refines a local rigidity result for the marked length spectrum of negatively curved manifolds, introduces a new pressure metric, and provides an alternative proof using geodesic stretch and statistical estimates.

Contribution

It offers an improved proof of marked length spectrum rigidity and introduces a novel pressure metric linking to Teichmüller theory.

Findings

Refined local rigidity result for marked length spectrum

Introduction of a new pressure metric on isometry classes

Connection of the pressure metric to Weil-Petersson metric

Abstract

We refine the recent local rigidity result for the marked length spectrum obtained by the first and third author in \cite{Guillarmou-Lefeuvre-18} and give an alternative proof using the geodesic stretch between two Anosov flows and some uniform estimate on the variance appearing in the central limit theorem for Anosov geodesic flows. In turn, we also introduce a new pressure metric on the space of isometry classes, that reduces to the Weil-Peterson metric in the case of Teichm\"uller space and is related to the works of \cite{McMullen,Bridgeman-Canary-Labourie-Sambarino-15}.