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 Hölder regularity -- described by a list ${\alpha }_1\le \cdots \le {\alpha }_{n-1}$ 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 Hölder potential.

19 pages, 5 figures