The comb representation of compact ultrametric spaces

We call a comb a map f : I → [0,∞), where I is a compact interval, such that {f ≥ ε} is finite for any ε. A comb induces a (pseudo)-distance d̄f on {f = 0} defined by d̄f (s, t) = max(s∧t,s∨t) f . We describe the completion Ī of {f = 0} for this metric, which is a compact ultrametric space called 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 . Running head. Comb metric spaces. 1 ar X iv :1 60 2. 08 24 6v 1 [ m at h. G N ] 2 6 Fe b 20 16 MSC 2000 subject classifications: Primary 05C05; secondary 46A19, 54E45, 54E70.