The comb representation of compact ultrametric spaces

TL;DR

This paper introduces a novel comb representation for compact ultrametric spaces, providing a unified way to model and analyze these spaces through a simple function-based construction.

Contribution

It establishes a one-to-one correspondence between compact ultrametric spaces and comb metric spaces, with explicit examples including well-known structures like the Kingman coalescent and $p$-adic fields.

Findings

Characterization of ultrametric spaces as comb metric spaces

Explicit construction of comb representations for classical ultrametric spaces

Application to spheres in real trees via contour processes

Abstract

We call a \emph{comb} a map $f:I\to [0,\infty)$, where $I$ is a compact interval, such that $\{f\ge \varepsilon\}$ is finite for any $\varepsilon$. A comb induces a (pseudo)-distance $\dtf$ on $\{f=0\}$ defined by $\dtf(s,t) = \max_{(s\wedge t, s\vee t)} f$. We describe the completion $\bar I$ of $\{f=0\}$ for this metric, which is a compact ultrametric space called \emph{comb metric space}. Conversely, we prove that any compact, ultrametric space $(U,d)$ without isolated points is isometric to a comb metric space. We show various examples of the comb representation of well-known ultrametric spaces: the Kingman coalescent, infinite sequences of a finite alphabet, the $p$-adic field and spheres of locally compact real trees. In particular, for a rooted, locally compact real tree defined from its contour process $h$, the comb isometric to the sphere of radius $T$ centered at the root can be extracted from $h$ as the depths of its excursions away from $T$.