Critical exponents and dimension for generalised limit sets

TL;DR

This paper explores the relationship between a newly defined critical exponent for discrete subsets of the unit ball and the fractal dimensions of their associated generalized limit sets, extending classical results in Kleinian group theory.

Contribution

It introduces a novel critical exponent for arbitrary discrete sets in the unit ball and studies its connection to fractal dimensions of generalized limit sets.

Findings

Established a link between the critical exponent and fractal dimensions

Extended classical Kleinian group results to more general discrete sets

Provided new insights into the structure of generalized limit sets

Abstract

There is a beautiful and well-studied relationship between the Poincare exponent and the fractal dimensions of the limit set of a Kleinian group. Motivated by this, given an arbitrary discrete subset of the unit ball we define a critical exponent and investigate how it relates to the fractal dimensions of the associated generalised limit set.