View article

A metric space (X, d) is called an ultrametric space if d satisfies the strong triangle inequality: d (x, y)≤ max {d (x, z), d (y, z)} for all x, y, z∈ X. Any compact perfect ultrametric space is homeomorphic to the ternary Cantor set C; it is quasisymmetric to C if and only if it is complete, doubling and uniformly perfect (see [4, Proposition 15.11]). It is a well-known fact that balls in ultrametric spaces possess some special properties not shared by balls in general metric spaces. For example,(1) balls are both open and closed;(2) any two balls are either disjoint or one is contained in the other;(3) every point in a ball can be its center;(4) every ball is a union of disjoint balls;(5) the diameter of a ball is less than or equal to its radius. These properties are easily derived from the strong triangle property (see, for example,[4, 14, 15]). Well-known examples of ultrametric spaces are the fields of p-adic numbers Qp, which are the object of study in p-adic Analysis and Fractal Geometry (see [10, 11, 12, 13, 16] and the references therein). A brief review of some selected topics in p-adic mathematical physics can be found in ([5]). Ultrametric spaces also arise as the boundaries at infinity of metric trees and more general Gromov 0-hyperbolic spaces (see, for instance,[2, 3, 8, 18]). Recently, Hughes has studied connections between metric trees and ultrametric spaces from a categorical point of view. He established an equivalence from the category of geodesically complete, rooted metric trees and the equivalence classes of isometries at infinity, to the category of complete ultrametric spaces of finite diameter and the local similarity equivalences ([8, Main Theorem]). The work …