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

Given a convex representation $\rho:\Gamma\to\textrm{PGL}(d,\mathbb{R})$ 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 H\"older 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 $\rho:\pi_1\Sigma\to\textrm{PSL}(d,\mathbb{R})$ is in the Hitchin component then $\alpha h_\rho\leq 2/(d-1)$ (where $\alpha$ is the H\"older exponent of the map $\zeta:\partial\mathbb{H}^2\to\mathscr{F}$) with equality if and only if $\rho$ is Fuchsian.