Abstract
We study various dimensions of spaces with nonpositive curvature in the A. D. Alexandrov sense, in particular, of ℝ-trees. We find some conditions necessary and sufficient for the metric space to be an ℝ-tree and clarify relations between the topological, Hausdorff, entropy, and rough dimensions. We build the examples of ℝ-trees and CAT(0)-spaces in which strict inequalities between the topological, Hausdorff, and entropy dimensions hold; we also show that the Hausdorff and entropy dimensions can be arbitrarily large while the topological dimension remains fixed.
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Dedicated to Yu. G. Reshetnyak on the occasion of his 75th birthday
Original Russian Text © P. D. Andreev and V. N. Berestovskiĭ, 2006, published in Matematicheskie Trudy, 2006, Vol. 9, No. 2, pp. 3–22.
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Andreev, P.D., Berestovskiĭ, V.N. Dimensions of ℝ-trees and self-similar fractal spaces of nonpositive curvature. Sib. Adv. Math. 17, 79–90 (2007). https://doi.org/10.3103/S1055134407020010
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DOI: https://doi.org/10.3103/S1055134407020010