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

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 $\rho$.

References in corpus (1)

• Entropy degeneration of convex projective surfaces

Cited by in corpus (4)

• The geometry of maximal representations of surface groups into SO(2,n)
• Optimal Bound on the Combinatorial Complexity of Approximating Polytopes
• Correlation of the renormalized Hilbert length for convex projective surfaces
• Proof of Kobayashi's rank conjecture on Clifford-Klein forms