Analyticity of Hausdorff dimension of limit sets of Kleinian groups

The Exponent of Convergence of Kleinian Groups; on a Theorem of Bishop and Jones

In this note we give a characterization of the Hausdorff dimensional significance of the exponent of convergence for any arbitrary Kleinian group. We show that this exponent is always equal to the Hausdorff dimension of the uniformly radial limit sets of the Kleinian group. We give a detailed and elementary proof of this important fact, clarifying and generalizing a result of Bishop and Jones. ...

Calculating Hausdorff Dimension of Julia Sets and Kleinian Limit Sets

Homological Dimension and Critical Exponent of Kleinian Groups

We prove an inequality between the relative homological dimension of a Kleinian group Γ ⊂ Isom(Hn) and its critical exponent. As an application of this result we show that for a geometrically finite Kleinian group Γ, if the topological dimension of the limit set of Γ equals its Hausdorff dimension, then the limit set is a round sphere.

Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups

This paper investigates the behavior of the Hausdorff dimensions of the limit sets Λn and Λ of a sequence of Kleinian groups Γn → Γ, where M = H/Γ is geometrically finite. We show if Γn → Γ strongly, then: (a) Mn = H 3/Γn is geometrically finite for all n ≫ 0, (b) Λn → Λ in the Hausdorff topology, and (c) H. dim(Λn) → H. dim(Λ), if H. dim(Λ) ≥ 1. On the other hand, we give examples showing the ...

Constructing Restricted Patterson Measures for Geometrically Infinite Kleinian Groups

In this paper, we study exhaustions, referred to as ρ-restrictions, of arbitrary nonelementary Kleinian groups with at most finitely many bounded parabolic elements. Special emphasis is put on the geometrically infinite case, where we obtain that the limit set of each of these Kleinian groups contains an infinite family of closed subsets, referred to as ρ-restricted limit sets, such that there ...