Regularity of Kleinian limit sets and Patterson-Sullivan measures

TL;DR

This paper investigates various notions of geometric regularity for Kleinian group limit sets and Patterson-Sullivan measures, revealing differences between these measures and classical fractal dimensions.

Contribution

It computes regularity and Assouad dimensions for limit sets and measures, highlighting their divergence from traditional fractal dimensions.

Findings

Assouad and lower dimensions are not always equal to the Poincaré exponent.

Computed upper and lower regularity dimensions of Patterson-Sullivan measures.

Analyzed local doubling properties of limit sets.

Abstract

We consider several (related) notions of geometric regularity in the context of limit sets of geometrically finite Kleinian groups and associated Patterson-Sullivan measures. We begin by computing the upper and lower regularity dimensions of the Patterson-Sullivan measure, which involves controlling the relative measure of concentric balls. We then compute the Assouad and lower dimensions of the limit set, which involves controlling local doubling properties. Unlike the Hausdorff, packing, and box-counting dimensions, we show that the Assouad and lower dimensions are not necessarily given by the Poincar\'e exponent.