The Hölder exponent of Anosov limit maps

Abstract

Let $\Gamma$ be a non-elementary word hyperbolic group and $d_a,a>1,$ a visual metric on its Gromov boundary ${\partial }_{\infty }\Gamma$. For an $1$-Anosov representation $\rho :\Gamma \to {\mathsf{GL}}_d(\mathbb{K})$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, we calculate the Hölder exponent of the Anosov limit map ${\xi }_{\rho }^1:({\partial }_{\infty }\Gamma ,d_a)\to (\mathbb{P}({\mathbb{K}}^d),d_{\mathbb{P}})$ of $\rho$ in terms of the moduli of eigenvalues of elements in $\rho (\Gamma )$ and the stable translation length on $\Gamma$. If $\rho$ is either irreducible or ${\xi }_{\rho }^1({\partial }_{\infty }\Gamma )$ spans ${\mathbb{K}}^d$ and $\rho$ is $\{1,2\}$-Anosov, then ${\xi }_{\rho }^1$ attains its Hölder exponent. We also provide an analogous calculation for the exponent of the inverse limit map of $(1,1,2)$-hyperconvex representations. Finally, we exhibit examples of non semisimple $1$-Anosov representations of surface groups in ${\mathsf{SL}}_4(\mathbb{R})$ whose Anosov limit map in $\mathbb{P}({\mathbb{R}}^4)$ does not attain its Hölder exponent.

arXiv comment26 pages, revisions made following referee's comments