Lipschitz images and dimensions
Rich\'ard Balka, Tam\'as Keleti

TL;DR
This paper investigates which compact metric spaces can be obtained as Lipschitz images of other spaces, establishing relationships between dimensions and mappings, and characterizing spaces via H"older images with connections to fractal dimensions.
Contribution
It provides new conditions relating Hausdorff and box dimensions for Lipschitz images, and characterizes spaces obtainable through H"older maps, advancing understanding of metric space mappings.
Findings
Lipschitz images exist when Hausdorff dimension exceeds box dimension.
Any natural dimension in is between Hausdorff and box dimensions.
Characterization of spaces as ta-Hf6lder images related to box dimension.
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 and are compact metric spaces and the Hausdorff dimension of is bigger than the upper box dimension of , then there exist a compact set and a Lipschitz onto map . As a corollary we prove that any `natural' dimension in must be between the Hausdorff and upper box dimensions. We show that if and are self-similar sets with the strong separation condition with equal Hausdorff dimension and is homogeneous, then can be mapped onto by a Lipschitz map if and only if and are bilipschitz equivalent. For given we also give a characterization of those…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Mathematical Dynamics and Fractals · Mathematical Analysis and Transform Methods
