On entropy, regularity and rigidity for convex representations of hyperbolic manifolds

Abstract

Given a convex representation $\rho :\Gamma \to \text{PGL}(d,\mathbb{R})$ of a convex co-compact group $\Gamma$ of ${\mathbb{H}}^k$ we find upper bounds for the quantity $\alpha h_{\rho },$ where $h_{\rho }$ is the entropy of $\rho$ and $\alpha$ is the Hölder exponent of the equivariant map $\partial \Gamma \to \mathbb{P}({\mathbb{R}}^d).$ We also give rigidity statements when the upper bound is attained. We then study Hitchin representations and prove that if $\rho :{\pi }_1\Sigma \to \text{PSL}(d,\mathbb{R})$ is in the Hitchin component then $\alpha h_{\rho }\le 2/(d-1)$ (where $\alpha$ is the Hölder exponent of the map $\zeta :\partial {\mathbb{H}}^2\to \mathcal{F}$) with equality if and only if $\rho$ is Fuchsian.

arXiv commentDue to discussions with Jean-Francois Quint, the paper has considerably shortened and improved