An invariant of bi-Lipschitz maps

A new numerical invariant for the category of compact metric spaces and Lipschitz maps is introduced. This invariant takes a value less than or equal to 1 for compact metric spaces that are Lipschitz isomorphic to ultrametric ones. Furthermore, a theorem is provided which makes it possible to compute this invariant for a large class of spaces. In particular, by utilizing this invariant, it is shown that neither a fat Cantor set nor the set ${\displaystyle \left\{0\right\}\cup {\left\{\frac{1}{n}\right\}}_{n\ge 1}}$ is Lipschitz isomorphic to an ultrametric space.