Explore Thurston's asymmetric metrics for Anosov representations in this 59-minute lecture by Leon Carvajales from Universidad de la Republica, Montevideo, presented at the Institut des Hautes Etudes Scientifiques (IHES). Begin with an overview of the Thurston metric, an asymmetric distance on the Teichmüller space of a surface, and its properties. Delve into the theory of Anosov representations, which generalizes aspects of classical Teichmüller-Thurston theory to higher rank representations of hyperbolic groups. Examine how Bridgeman-Canary-Labourie-Sambarino applied Thermodynamical Formalism to construct pressure metrics on spaces of Anosov representations, generalizing the Weil-Petersson metric. Learn about the application of similar techniques to show how Thurston's asymmetric distance extends to this setting, based on joint work with Xian Dai, Beatrice Pozzetti, and Anna Wienhard.