Entropy of Hilbert metrics and length spectrum of Hitchin representations in $\mathrm{PSL}(3,\mathbb{R})$

TL;DR

This paper establishes a sharp inequality between Blaschke and Hilbert distances in convex domains, leading to volume entropy rigidity in Hilbert geometries and a comparison of length spectra for Hitchin representations in PSL(3,R).

Contribution

It proves a new inequality relating Blaschke and Hilbert distances, with implications for volume entropy rigidity and length spectrum comparison in Hitchin representations.

Findings

Proves the inequality d^B(x,y) < d^H(x,y) + 1 for convex domains.

Establishes volume entropy growth bound of e^{(n-1)R} in Hilbert geometries.

Shows existence of a Fuchsian representation with uniformly smaller length spectrum than any Hitchin representation.

Abstract

We prove a sharp inequality between the Blaschke and Hilbert distance on a proper convex domain: for any two points $x$ and $y$, \[d^B(x,y) < d^H(x,y) +1.\] We obtain two interesting consequences: the first one is the volume entropy rigidity for Hilbert geometries : for any proper convex domain of $\mathbb{R}\mathbf{P}^n$, the volume of a ball of radius $R$ grows at most like $e^{(n-1)R}$. The second consequence is the following fact: for any Hitchin representation of a surface group into $\mathrm{PSL}(3,\mathbb{R})$, there exists a Fuchsian representation $j$ in $\mathrm{PSL}(2,\mathbb{R})$ such that the length spectrum of $j$ is uniformly smaller than the length spectrum of $\rho$.