Simplicity of Lyapunov spectra and boundaries of non-conical strictly convex divisible sets

TL;DR

This paper proves that the boundary regularity of certain convex projective sets is maximally anisotropic, which is linked to the simplicity of the Lyapunov spectrum of the associated Hilbert geodesic flow.

Contribution

It establishes the maximal anisotropy of boundary regularity for non-ellipsoidal convex divisible sets, connecting it to the simplicity of the Lyapunov spectrum.

Findings

Boundary regularity is maximally anisotropic with no repeated regularity exponents.

The Lyapunov spectrum of the Hilbert geodesic flow is simple for all equilibrium measures.

The result applies to strictly convex divisible sets that are not ellipsoids.

Abstract

Let $\Omega$ be a strictly convex divisible subset of the $n$-dimensional real projective space which is not an ellipsoid. Even though $\partial\Omega$ is not $C^2$, Benoist showed that it is $C^{1+\alpha}$ for some $\alpha>0$, and Crampon established that $\partial\Omega$ actually possesses a sort of anisotropic H\"older regularity -- described by a list $\alpha_1\leq\dots\leq\alpha_{n-1}$ of positive real numbers -- at almost all of its points. In this article, we show that $\partial\Omega$ is maximally anisotropic in the sense that this list of approximate regularities of $\partial\Omega$ does not contain repetitions. This result is a consequence of the simplicity of the Lyapunov spectrum of the Hilbert geodesic flow for every equilibrium measure associated to a H\"older potential.