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| Format: | Preprint |
| Publié: |
2023
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| Accès en ligne: | https://arxiv.org/abs/2306.15823 |
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| _version_ | 1866909737169190912 |
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| author | Tsouvalas, Konstantinos |
| author_facet | Tsouvalas, Konstantinos |
| contents | Let $Γ$ be a non-elementary word hyperbolic group and $d_{a}, a>1,$ a visual metric on its Gromov boundary $\partial_{\infty}Γ$. For an $1$-Anosov representation $ρ:Γ\rightarrow \mathsf{GL}_{d}(\mathbb{K})$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, we calculate the Hölder exponent of the Anosov limit map $ξ_ρ^1:(\partial_{\infty}Γ, d_{a})\rightarrow (\mathbb{P}(\mathbb{K}^d),d_{\mathbb{P}})$ of $ρ$ in terms of the moduli of eigenvalues of elements in $ρ(Γ)$ and the stable translation length on $Γ$. If $ρ$ is either irreducible or $ξ_ρ^1(\partial_{\infty}Γ)$ spans $\mathbb{K}^d$ and $ρ$ is $\{1,2\}$-Anosov, then $ξ_ρ^1$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15823 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Hölder exponent of Anosov limit maps Tsouvalas, Konstantinos Dynamical Systems Group Theory Let $Γ$ be a non-elementary word hyperbolic group and $d_{a}, a>1,$ a visual metric on its Gromov boundary $\partial_{\infty}Γ$. For an $1$-Anosov representation $ρ:Γ\rightarrow \mathsf{GL}_{d}(\mathbb{K})$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, we calculate the Hölder exponent of the Anosov limit map $ξ_ρ^1:(\partial_{\infty}Γ, d_{a})\rightarrow (\mathbb{P}(\mathbb{K}^d),d_{\mathbb{P}})$ of $ρ$ in terms of the moduli of eigenvalues of elements in $ρ(Γ)$ and the stable translation length on $Γ$. If $ρ$ is either irreducible or $ξ_ρ^1(\partial_{\infty}Γ)$ spans $\mathbb{K}^d$ and $ρ$ is $\{1,2\}$-Anosov, then $ξ_ρ^1$ 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. |
| title | The Hölder exponent of Anosov limit maps |
| topic | Dynamical Systems Group Theory |
| url | https://arxiv.org/abs/2306.15823 |