Checking atomicity of conformal ending measures for Kleinian groups

TL;DR

This paper investigates the conditions under which conformal ending measures for Kleinian groups are purely atomic, showing that atoms in the big horospherical limit set must be parabolic fixed points, with detailed examples provided.

Contribution

It establishes sufficient conditions for atomicity of conformal ending measures and characterizes atoms within the big horospherical limit set for Kleinian groups.

Findings

Atoms in the big horospherical limit set are parabolic fixed points.

Provided examples of purely atomic and non-atomic conformal ending measures.

Identified conditions ensuring atomicity of measures.

Abstract

In this paper we address questions of continuity and atomicity of conformal ending measures for arbitrary non-elementary Kleinian groups. We give sufficient conditions under which such ending measures are purely atomic. Moreover, we will show that if a conformal ending measure has an atom which is contained in the big horospherical limit set, then this atom has to be a parabolic fixed point. Also, we give detailed discussions of non-trivial examples for purely atomic as well as for non-atomic conformal ending measures.