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

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.