Conformality for a robust class of non-conformal attractors

Abstract

In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that the Hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent.

Entropy rigidity for cusped Hitchin representations

“…We obtain upper bounds on the Hausdorff dimensions of conical limit sets of P k -transverse groups, generalizing results of Glorieux-Montclair-Tholozan [27] and Pozzetti-Sambarino-Wienhard [43] from the Anosov setting.…”

“…We show that for such subgroups (with σ 2 (γ) = σ q (γ) for all γ ∈ Γ), the Hausdorff dimension of the conical limit set agrees with the first simple root critical exponent. This result is a common generalization of results of Pozzetti-Sambarino-Wienhard [43], for Anosov groups, and Bishop-Jones [6], for Kleinian groups, and the proof makes use of techniques drawn from each source. We observe that the φ-entropy and the φ-critical exponent of a geometrically finite Hitchin representation agree.…”

“…We further introduce the class of (1, 1, q)-hypertransverse subgroups which include images of the (1, 1, q)-hyperconvex Anosov representations introduced by Pozzetti-Sambarino-Wienhard [43], their subgroups, Hitchin representations of Fuchsian groups, and all discrete subgroups of PO(d − 1, 1). We show that for such subgroups (with σ 2 (γ) = σ q (γ) for all γ ∈ Γ), the Hausdorff dimension of the conical limit set agrees with the first simple root critical exponent.…”

Topological restrictions on Anosov representations

“…Note that if , then ‐hyperconvex representations are ‐hyperconvex, so we may conclude that if is ‐hyperconvex, then has cohomological dimension at most . Pozzetti, Sambarino, and Wienhard [45, Corollary 7.6] observe that if and , then the symmetric power is ‐hyperconvex, so one obtains no additional topological restrictions in the case where is ‐hyperconvex and has maximal cohomological dimension .…”

“…A representation is said to be ‐hyperconvex , where , if is , , and (or )‐Anosov and whenever are distinct, One may view the following as a generalization of Corollary 6.6 of Pozzetti–Sambarino–Wienhard [45] which asserts that if is ‐hyperconvex and , then there is a continuous injection of into , see also Lemma 4.10 in Zhang–Zimmer [50]. (Pozzetti, Sambarino, and Wienhard's result [45, Corollary 6.6] also applies to representations into where is any loc...…”