Bounded distortion homeomorphisms on ultrametric spaces

TL;DR

This paper characterizes bounded distortion homeomorphisms on ultrametric spaces derived from bushy trees, revealing their unique properties and distinctions from quasiconformal and bi-Hölder mappings.

Contribution

It introduces a bounded distortion property that characterizes power quasi-symmetric homeomorphisms on certain ultrametric spaces, expanding understanding of their geometric structure.

Findings

Bounded distortion characterizes power quasi-symmetric homeomorphisms.

Examples show differences from quasiconformal and bi-Hölder conditions.

Ultrametric spaces from bushy trees are key to these properties.

Abstract

It is well-known that quasi-isometries between R-trees induce power quasi-symmetric homeomorphisms between their ultrametric end spaces. This paper investigates power quasi-symmetric homeomorphisms between bounded, complete, uniformly perfect, ultrametric spaces (i.e., those ultrametric spaces arising up to similarity as the end spaces of bushy trees). A bounded distortion property is found that characterizes power quasi-symmetric homeomorphisms between such ultrametric spaces that are also pseudo-doubling. Moreover, examples are given showing the extent to which the power quasi-symmetry of homeomorphisms is not captured by the quasiconformal and bi-H\"older conditions for this class of ultrametric spaces.