The pressure metric for Anosov representations

Abstract

Using the thermodynamic formalism, we introduce a notion of intersection for projective Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce an Out $${(\Gamma)}$$ -invariant Riemannian metric on the smooth points of the deformation space of irreducible, generic, projective Anosov representations of a word hyperbolic group $${\Gamma}$$ into $${\mathsf{SL}_m(\mathbb{R})}$$ . In particular, we produce mapping class group invariant Riemannian metrics on Hitchin components which restrict to the Weil–Petersson metric on the Fuchsian loci. Moreover, we produce $${{\rm Out}(\Gamma)}$$ -invariant metrics on deformation spaces of convex cocompact representations into $${\mathsf{PSL}_2(\mathbb{C})}$$ and show that the Hausdorff dimension of the limit set varies analytically over analytic families of convex cocompact representations into any rank 1 semi-simple Lie group.