The Hölder exponent of Anosov limit maps

2023/06/27 by Konstantinos Tsouvalas, Tsouvalas, Konstantinos · 2 citations
Computer Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2306.15823

openalex publication_date 2023/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Γ be a non-elementary word hyperbolic group and d_a, a>1, a visual metric on its Gromov boundary ∂_∞Γ. For an 1-Anosov representation ρ:Γ→ GL_d(\mathbbK), where \mathbbK=ℝ or ℂ, we calculate the Hölder exponent of the Anosov limit map ξ_ρ¹:(∂_∞Γ, d_a)→ (ℙ(\mathbbK^d),d_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_ρ¹(∂_∞Γ) spans \mathbbK^d and ρ is \1,2\-Anosov, then ξ_ρ¹ 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 SL₄(ℝ) whose Anosov limit map in ℙ(ℝ⁴) does not attain its Hölder exponent.

Cited by

• Flexibility and degeneracy around a theorem of Thurston
• A simultaneous Abels-Margulis-Soifer lemma

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