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

Given a convex representation $ρ:Γ\to\textrm{PGL}(d,\mathbb{R})$ of a convex co-compact group $Γ$ of $\mathbb{H}^k$ we find upper bounds for the quantity $αh_ρ,$ where $h_ρ$ is the entropy of $ρ$ and $α$ is the Hölder exponent of the equivariant map $\partialΓ\to\mathbb{P}(\mathbb{R}^d).$ We also give rigidity statements when the upper bound is attained. We then study Hitchin representations and prove that if $ρ:π_1Σ\to\textrm{PSL}(d,\mathbb{R})$ is in the Hitchin component then $αh_ρ\leq 2/(d-1)$ (where $α$ is the Hölder exponent of the map $ζ:\partial\mathbb{H}^2\to\mathscr{F}$) with equality if and only if $ρ$ is Fuchsian.

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