The Hölder exponent of Anosov limit maps

• Published in: arXiv.org 2023-06
• Main Author: Tsouvalas, Konstantinos
• Format: Article
• Language: English
• Subjects: Eigenvalues; Mathematical analysis; Representations
• ISSN: 2331-8422

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.