Appears in collection : Homogeneous Dynamics and Geometry in Higher-Rank Lie Groups

The Thurston metric is an asymmetric distance on the Teichmüller space of a surface, which is computed by comparing the lengths of closed curves in the two hyperbolic structures. Thurston introduced this metric and proved many interesting properties of it, which we will briefly summarize. The theory of Anosov representations aims to generalize several aspects of the classical Teichmüller-Thurston theory to higher rank representations of hyperbolic groups. For instance, Bridgeman-Canary-Labourie-Sambarino applied the Thermodynamical Formalism to the underlying geodesic flow to construct pressure metrics on some spaces of Anosov representations, which generalize the Weil-Petersson metric on Teichmüller space. In this talk we will apply similar techniques to show that Thurston's asymmetric distance also generalizes to this setting. This is joint work with Xian Dai, Beatrice Pozzetti and Anna Wienhard.