The H\"older exponent of Anosov limit maps

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 $\rho:\Gamma \rightarrow \mathsf{GL}_{d}(\mathbb{K})$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, we calculate the H\"older exponent of the Anosov limit map $\xi_{\rho}^1:(\partial_{\infty}\Gamma, d_{a})\rightarrow (\mathbb{P}(\mathbb{K}^d),d_{\mathbb{P}})$ 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 H\"older 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 H\"older exponent.