Thermodynamic Formalism on SRB Measures of Compact Convex Projective manifolds, Geometry and Counting

We study the Hilbert geodesic flow on a closed convex real projective $n$-dimensional manifold. We use the thermodynamic formalism for this dissipative Anosov flow to obtain a geometric description of the SRB measure. We also deduce new counting properties for the induced representations of the fundamental groups in $SL(n+1,\mathbb{R})$.