Some invariant properties of quasi-M\"obius maps

TL;DR

This paper studies properties of quasi-M"obius maps on quasi-metric spaces, demonstrating that doubling and uniform disconnectedness are invariant under these maps, which enhances understanding of their geometric structure.

Contribution

It proves that the doubling property and uniform disconnectedness are invariant under quasi-M"obius maps in quasi-metric spaces, extending known invariance results.

Findings

Doubling property is invariant under quasi-M"obius maps.

Uniform disconnectedness is invariant under quasi-M"obius maps.

Provides new insights into geometric invariants of quasi-M"obius spaces.

Abstract

We investigate properties which remain invariant under the action of quasi-M\"obius maps of quasi-metric spaces. A metric space is called doubling with constant D if every ball of finite radius can be covered by at most D balls of half the radius. It is shown that the doubling property is an invariant property for (quasi-)M\"obius spaces. Additionally it is shown that the property of uniform disconnectedness is an invariant for (quasi-)M\"obius spaces as well.