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

TL;DR

This paper establishes bounds on the product of entropy and H"older exponent for convex representations of hyperbolic groups, providing rigidity results and characterizing when equality holds, especially for Hitchin representations.

Contribution

It introduces new upper bounds for entropy-H"older product in convex representations and characterizes cases of equality, including for Hitchin representations.

Findings

Upper bounds for lpha h_ ho are derived.

Rigidity results when bounds are attained.

Characterization of Fuchsian representations in Hitchin component.

Abstract

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.