Hausdorff Dimension of Anosov Subgroups' Limit Sets with Special Self-Affine Complexity

Abstract

Let $Γ\subset \mathsf{PGL}(d,\mathbb{R})$ be an irreducible projective Anosov subgroup and let $Λ^1(Γ)$ be its projective limit set. Viewing $Λ^1(Γ)$ as an analogue of a self-affine set, we investigate the Hausdorff dimension of $Λ^1(Γ)$ under specific assumptions regarding its affine complexity: 1. If $Λ^1(Γ)$ is of full Hausdorff dimension, then $d= 2$ and $Γ$ is a cocompact lattice. 2. If $d = 3$ and $Γ$ is the image of a closed surface group under an irreducible Anosov representation, then $Λ^1(Γ)$ never has Hausdorff dimension $1$ unless the representation is Hitchin. 3. If the limit set $Λ^1(Γ)$ exhibits a partial quasi-self-similarity (in the sense of Falconer~\cite{falconerselfsimilar1}) -- which can be implied by the ``regular distortion property'' of $Γ$ -- then the Hausdorff dimension of $Λ^1(Γ)$ equals the critical exponent of the first simple root. An application of this result is the computation of the Hausdorff dimension of the limit set for arbitrary $Θ$-positive representations of convex cocompact Fuchsian groups.