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Ultrametrics and geometric measures

Authors: H. Movahedi-Lankarani and R. Wells
Journal: Proc. Amer. Math. Soc. 123 (1995), 2579-2584
MSC: Primary 54E40; Secondary 28A75, 54E35
MathSciNet review: 1307552
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Abstract: Let Z be a locally connected, locally compact, and separable metric space equipped with a geometric measure v. It is shown that if a subset Y of Z is bi-Lipschitz isomorphic to an ultrametric space, then $ \nu (Y) = 0$.

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  • [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. MR 918059 (89d:54020)
  • [2] P. Assouad, Étude d'une dimension métrique liée à la possibilité de plongements dans $ {\mathbb{R}^N}$, C. R. Acad. Sci. Paris Sér. A 288 (1979), 731-734. MR 532401 (80f:54030)
  • [3] A. Jonsson and H. Wallin, Function spaces on subsets of $ {\mathbb{R}^N}$, Math. Rep. (Chur, Switzerland) (J. Peetre, ed.), vol. 2, pt. 1, Harwood Academic, New York, 1984. MR 820626 (87f:46056)
  • [4] J. B. Kelly, Metric inequalities and symmetric differences, Inequalities-II (O. Shisha, ed.), Academic Press, New York, 1970, pp. 193-212. MR 0264600 (41:9192)
  • [5] A. Yu. Lemin, Isometric imbedding of isosceles (non-Archimedean) spaces in Euclidean spaces, Dokl. Akad. Nauk SSSR 285 (1985), 558-562; English transl., Soviet Math. Dokl. 32 (1985), 740-744. MR 821340 (87h:54056)
  • [6] K. Luosto, private communication.
  • [7] J. Luukkainen and H. Movahedi-Lankarani, Minimal bi-Lipschitz embedding dimension of ultrametric spaces, Fund. Math. 144 (1994), 181-193. MR 1273695 (95i:54031)
  • [8] H. Movahedi-Lankarani, An invariant of bi-Lipschitz maps, Fund. Math. 143 (1993), 1-9. MR 1234987 (94i:54063)
  • [9] S. Semmes, private communication.
  • [10] A. F. Timan, On the isometric mapping of some ultrametric spaces into $ {L_p}$-spaces, Trudy Mat. Inst. Steklov. 134 (1975), 314-326; English transl., Proc. Steklov Inst. Math. 134 (1975), 357-370. MR 0390618 (52:11443)
  • [11] A. F. Timan and I. A. Vestfrid, Any separable ultrametric space can be isometrically imbedded in $ {\ell _2}$, Funktsional. Anal. i Prilozhen 17 (1983), no. 1, 85-86; English transl., Functional Anal. Appl 17 (1983), 70-71. MR 695109 (85b:54048)

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Keywords: Lipschitz, ultrametric, geometric measure, Lebesgue measure, logarithmic ratio, Hausdorff dimension, metric dimension
Article copyright: © Copyright 1995 American Mathematical Society

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