The Hölder exponent of Anosov limit maps

Let $\mathrm{\Gamma }$ be a non-elementary word hyperbolic group and $d_a,a>1,$ a visual metric on its Gromov boundary ${\mathrm{\partial }}_{\mathrm{\infty }}\mathrm{\Gamma }$. For an $1$-Anosov representation $\rho :\mathrm{\Gamma }\to {\mathsf{G}\mathsf{L}}_d(\mathbb{K})$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, we calculate the Hölder exponent of the Anosov limit map ${\xi }_{\rho }^1:({\mathrm{\partial }}_{\mathrm{\infty }}\mathrm{\Gamma },d_a)\to (\mathbb{P}({\mathbb{K}}^d),d_{\mathbb{P}})$ of $\rho$ in terms of the moduli of eigenvalues of elements in $\rho (\mathrm{\Gamma })$ and the stable translation length on $\mathrm{\Gamma }$. If $\rho$ is either irreducible or ${\xi }_{\rho }^1({\mathrm{\partial }}_{\mathrm{\infty }}\mathrm{\Gamma })$ 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 ${\mathsf{S}\mathsf{L}}_4(\mathbb{R})$ whose Anosov limit map in $\mathbb{P}({\mathbb{R}}^4)$ does not attain its Hölder exponent.