Lipschitz images and dimensions

Balka, Richárd and Keleti, Tamás (2024) Lipschitz images and dimensions. ADVANCES IN MATHEMATICS, 446. No. 109669. ISSN 0001-8708

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Abstract

We consider the question which compact metric spaces can be obtained as a Lipschitz image of the middle third Cantor set, or more generally, as a Lipschitz image of a subset of a given compact metric space. In the general case we prove that if A and B are compact metric spaces and the Hausdorff dimension of A is bigger than the upper box dimension of B, then there exist a compact set A ⊂ A and a Lipschitz onto map f : A → B. As a corollary we prove that any ‘natural’ dimension in Rn must be between the Hausdorff and upper box dimensions. We show that if A and B are self-similar sets with the strong separation condition with equal Hausdorff dimension and A is homogeneous, then A can be mapped onto B by a Lipschitz map if and only if A and B are bilipschitz equivalent. For given α > 0 we also give a characterization of those compact metric spaces that can be obtained as an α-Hölder image of a compact subset of R. The quantity we introduce for this turns out to be closely related to the upper box dimension.

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