Minimal bi-Lipschitz embedding dimension of ultrametric spaces

Jouni Luukkainen; Hossein Movahedi-Lankarani

Fundamenta Mathematicae (1994)

  • Volume: 144, Issue: 2, page 181-193
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Luukkainen, Jouni, and Movahedi-Lankarani, Hossein. "Minimal bi-Lipschitz embedding dimension of ultrametric spaces." Fundamenta Mathematicae 144.2 (1994): 181-193. <http://eudml.org/doc/212022>.

@article{Luukkainen1994,
abstract = {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.},
author = {Luukkainen, Jouni, Movahedi-Lankarani, Hossein},
journal = {Fundamenta Mathematicae},
keywords = {bi-Lipschitz; ultrametric; metric dimension; inverse limit},
language = {eng},
number = {2},
pages = {181-193},
title = {Minimal bi-Lipschitz embedding dimension of ultrametric spaces},
url = {http://eudml.org/doc/212022},
volume = {144},
year = {1994},
}

TY - JOUR
AU - Luukkainen, Jouni
AU - Movahedi-Lankarani, Hossein
TI - Minimal bi-Lipschitz embedding dimension of ultrametric spaces
JO - Fundamenta Mathematicae
PY - 1994
VL - 144
IS - 2
SP - 181
EP - 193
AB - 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.
LA - eng
KW - bi-Lipschitz; ultrametric; metric dimension; inverse limit
UR - http://eudml.org/doc/212022
ER -

References

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  1. [1] M. Aschbacher, P. Baldi, E. B. Baum and R. M. Wilson, Embeddings of ultrametric spaces in finite dimensional structures, SIAM J. Algebraic Discrete Methods 8 (1987), 564-577. Zbl0639.51018
  2. [2] P. Assouad, Espaces métriques, plongements, facteurs, Thèse de doctorat d'État, Orsay, 1977. Zbl0396.46035
  3. [3] P. Assouad, Étude d’une dimension métrique liée à la possibilité de plongements dans ℝ n , C. R. Acad. Sci. Paris Sér. A 288 (1979), 731-734. Zbl0409.54020
  4. [4] P. Assouad, Plongements Lipschitziens dans ℝ n , Bull. Soc. Math. France 111 (1983), 429-448. Zbl0597.54015
  5. [5] A. Ben-Artzi, A. Eden, C. Foias and B. Nicolaenko, Hölder continuity for the inverse of Ma né's projection, J. Math. Anal. Appl. 178 (1993), 22-29. Zbl0815.46016
  6. [6] R. Engelking, Dimension Theory, PWN, Warszawa, and North-Holland, Amsterdam, 1978. 
  7. [7] K. Falconer, Fractal Geometry, Wiley, Chichester, 1990. 
  8. [8] J. B. Kelly, Metric inequalities and symmetric differences, in: Inequalities-II, O. Shisha (ed.), Academic Press, New York, 1970, 193-212. 
  9. [9] J. B. Kelly, Hypermetric spaces and metric transforms, in: Inequalities-III, O. Shisha (ed.), Academic Press, New York, 1972, 149-158. Zbl0294.52011
  10. [10] A. Yu. Lemin, On the stability of the property of a space being isosceles, Uspekhi Mat. Nauk 39 (5) (1984), 249-250 (in Russian); English transl.: Russian Math. Surveys 39 (5) (1984), 283-284. 
  11. [11] A. Yu. Lemin, Isometric imbedding of isosceles (non-Archimedean) spaces in Euclidean spaces, Dokl. Akad. Nauk SSSR 285 (1985), 558-562 (in Russian); English transl.: Soviet Math. Dokl. 32 (1985), 740-744. Zbl0602.54031
  12. [12] J. Luukkainen and P. Tukia, Quasisymmetric and Lipschitz approximation of embeddings, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 343-367. Zbl0456.57005
  13. [13] J. Luukkainen and J. Väisälä, Elements of Lipschitz topology, ibid. 3 (1977), 85-122. Zbl0397.57011
  14. [14] G. Michon, Les cantors réguliers, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), 673-675. Zbl0582.54019
  15. [15] H. Movahedi-Lankarani, On the inverse of Ma né's projection, Proc. Amer. Math. Soc. 116 (1992), 555-560. Zbl0777.54011
  16. [16] H. Movahedi-Lankarani, An invariant of bi-Lipschitz maps, Fund. Math. 143 (1993), 1-9. Zbl0845.54018
  17. [17] A. C. M. van Rooij, Non-Archimedean Functional Analysis, Marcel Dekker, New York, 1978. 
  18. [18] W. H. Schikhof, Ultrametric Calculus, Cambridge University Press, Cambridge, 1984. Zbl0553.26006
  19. [19] A. F. Timan, On the isometric mapping of some ultrametric spaces into L p -spaces, Trudy Mat. Inst. Steklov. 134 (1975), 314-326 (in Russian); English transl.: Proc. Steklov Inst. Math. 134 (1975), 357-370. Zbl0329.46031
  20. [20] A. F. Timan and I. A. Vestfrid, Any separable ultrametric space can be isometrically imbedded in l 2 , Funktsional. Anal. i Prilozhen. 17 (1) (1983), 85-86 (in Russian); English transl.: Functional Anal. Appl. 17 (1983), 70-71. Zbl0522.46017
  21. [21] J. H. Wells and L. R. Williams, Embeddings and Extensions in Analysis, Springer, Berlin, 1975. Zbl0324.46034

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