The comb representation of compact ultrametric spaces

Abstract

We call a \emph{comb} a map $f:I\to [0,\infty )$, where $I$ is a compact interval, such that $\{f\ge \epsilon \}$ is finite for any $\epsilon$. 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 $T$.

24 pages, 3 figures

References in corpus (2)

• Asymptotic genealogy of a critical branching process
• The comb-like representations of cellular ordinal balleans

Cited by in corpus (3)

• Mutations on a Random Binary Tree with Measured Boundary
• On quasisymmetric mappings between ultrametric spaces
• Minimum spanning paths and Hausdorff distance in finite ultrametric spaces