The comb representation of compact ultrametric spaces

Abstract

We call a comb a map f: I → [0,∞), where I is a compact interval, such that {f ≥ ε} is finite for any ε > 0. A comb induces a (pseudo)-distance \({\overline d _f}\) on {f = 0} defined by \({\overline d _f}\left( {s,t} \right) = {\max _{\left( {s \wedge t,s \vee t} \right)}}f\). We describe the completion \(\overline I \) of {f = 0} for this metric, which is a compact ultrametric space called the 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.