Ultrametric spaces and the logarithmic ratio

TL;DR

This paper investigates the relationship between ultrametric spaces and the logarithmic ratio, establishing conditions under which a space is bi-Hölder equivalent to an ultrametric space and constructing examples with prescribed logarithmic ratios.

Contribution

It provides new characterizations of ultrametric spaces via the finiteness of the logarithmic ratio and constructs examples with arbitrary logarithmic ratios.

Findings

Finiteness of R(X,d) implies bi-Hölder equivalence to ultrametric spaces.

Existence of spaces with any prescribed logarithmic ratio s.

A bi-Hölder embedding theorem for certain totally disconnected spaces.

Abstract

It is shown that if a compact metric space $(X, d)$ is bi-H\"older equivalent to an ultrametric space, then the logarithmic ratio $R(X,d)$ is finite. Conversely, if the logarithmic ratio $R(X,d)$ is finite and ${\A}^*_p (X) \ne \emptyset$ for some $p \in (1, \infty )$, then $(X, d)$ is bi-H\"older equivalent to an ultrametric space. It is also shown that for any $s \in [0, \infty]$ there exists a compact countable metric space $(X, d)$ with a unique cluster point such that the logarithmic ratio $R(X, d)$ is equal $s$. Moreover, we prove a bi-H\"older embedding result for a certain class of compact totally disconnected metric spaces.