Geodesic stretch, pressure metric and marked length spectrum rigidity

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