Ultrametric spaces bi-Lipschitz embeddable in $ℝ^n$

• Autorzy: Kerkko Luosto
• Języki publikacji: EN

Abstrakty

EN

It is proved that if an ultrametric space can be bi-Lipschitz embedded in $ℝ^n$, then its Assouad dimension is less than n. Together with a result of Luukkainen and Movahedi-Lankarani, where the converse was shown, this gives a characterization in terms of Assouad dimension of the ultrametric spaces which are bi-Lipschitz embeddable in $ℝ^n$.

Kategorie tematyczne

• 54F45: Dimension theory
• 54E40: Special maps on metric spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1996
• Tom: 150
• Numer: 1
• Strony: 25-42

Daty

wydano

1996

otrzymano

1995-06-07

poprawiono

1996-01-09

Twórcy

autor

Kerkko Luosto

• Department of Mathematics, University of Helsinki, P.O. Box 4 (Yliopistonkatu 5), FIN-00014 Helsinki, Finland

Bibliografia

• [ABBW] 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.
• [A] 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.
• [LM-L] J. Luukkainen and H. Movahedi-Lankarani, Minimal bi-Lipschitz embedding dimension of ultrametric spaces, Fund. Math. 144 (1994), 181-193.
• [M-LW] H. Movahedi-Lankarani and R. Wells, Ultrametrics and geometric measures, Proc. Amer. Math. Soc. 123 (1995), 2579-2584.
• [S] S. Semmes, On the nonexistence of bilipschitz parameterizations and geometric problems about $A_∞$ weights, Rev. Mat. Iberoamericana, to appear.