On the limit set of a geometrically infinite Kleinian group

TL;DR

This paper investigates the structure of the limit set of a geometrically infinite Kleinian group, analyzing measure-theoretic properties and decompositions, to deepen understanding of Kleinian group dynamics.

Contribution

It provides a comparison between Patterson-Sullivan measure and harmonic measure on the limit set of such groups, especially when the limit set equals the sphere at infinity.

Findings

Decomposition of the limit set into conical and ending limit sets.

Comparison of Patterson-Sullivan and harmonic measures.

Results specific to groups with limit set equal to the sphere at infinity.

Abstract

For a torsion free Kleinian group $\Gamma$ without parabolics, we consider the decomposition of the limit set $L(\Gamma)$ into conical and ending limit sets and compare the Patterson-Sullivan measure with the harmonic measure on $L(\Gamma)$ when $L(\Gamma)=S^2_\infty$.