An invariant of bi-Lipschitz maps

Hossein Movahedi-Lankarani

Fundamenta Mathematicae (1993)

  • Volume: 143, Issue: 1, page 1-9
  • ISSN: 0016-2736

Abstract

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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 ∪ 1 / n n ≥ 1 is Lipschitz isomorphic to an ultrametric space.

How to cite

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Movahedi-Lankarani, Hossein. "An invariant of bi-Lipschitz maps." Fundamenta Mathematicae 143.1 (1993): 1-9. <http://eudml.org/doc/211989>.

@article{Movahedi1993,
abstract = {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 $\{0\}∪\{1/n\}_\{n≥1\}$ is Lipschitz isomorphic to an ultrametric space.},
author = {Movahedi-Lankarani, Hossein},
journal = {Fundamenta Mathematicae},
keywords = {invariant for the category of compact metric spaces; Lipschitz maps; Lipschitz; ultrametric space},
language = {eng},
number = {1},
pages = {1-9},
title = {An invariant of bi-Lipschitz maps},
url = {http://eudml.org/doc/211989},
volume = {143},
year = {1993},
}

TY - JOUR
AU - Movahedi-Lankarani, Hossein
TI - An invariant of bi-Lipschitz maps
JO - Fundamenta Mathematicae
PY - 1993
VL - 143
IS - 1
SP - 1
EP - 9
AB - 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 ${0}∪{1/n}_{n≥1}$ is Lipschitz isomorphic to an ultrametric space.
LA - eng
KW - invariant for the category of compact metric spaces; Lipschitz maps; Lipschitz; ultrametric space
UR - http://eudml.org/doc/211989
ER -

References

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  1. [1] G. Michon, Les cantors réguliers, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), 673-675. Zbl0582.54019
  2. [2] G. Michon, Le théorème de Frostman pour les ensembles de Cantor réguliers, ibid. 305 (1987), 265-268. Zbl0629.28002
  3. [3] G. Michon, Applications du théorème de Frostman à la dimension des ensembles de Cantor réguliers, ibid. 305 (1987), 689-692. Zbl0654.28004
  4. [4] D. Sullivan, Differentiable structures on fractal-like sets, determined by intrinsic scaling functions on dual Cantor sets, in: The Mathematical Heritage of Hermann Weyl, R. O. Wells, Jr. (ed.), Proc. Sympos. Pure Math. 48, Amer. Math. Soc., Providence, R.I., 1988, 15-23. 

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