Lipschitz images and dimensions

2023/08/04 by Balka, Richárd, Keleti, Tamás · 2 citations
#28A78 #28A80 #51F30 #54E45 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2308.02639

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\colon 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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