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An invariant of bi-Lipschitz maps

Hossein Movahedi-Lankarani — 1993

Fundamenta Mathematicae

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 $0 \cup {1 / n}_{n \geq 1}$ is Lipschitz isomorphic to an ultrametric space.

Minimal bi-Lipschitz embedding dimension of ultrametric spaces

Jouni Luukkainen; Hossein Movahedi-Lankarani — 1994

Fundamenta Mathematicae

We prove that an ultrametric space can be bi-Lipschitz embedded in $ℝ^{n}$ if its metric dimension in Assouad’s sense is smaller than n. We also characterize ultrametric spaces up to bi-Lipschitz homeomorphism as dense subspaces of ultrametric inverse limits of certain inverse sequences of discrete spaces.

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