Lipschitz images and dimensions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913305912672256 |
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| author | Balka, Richárd Keleti, Tamás |
| author_facet | Balka, Richárd Keleti, Tamás |
| contents | 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 $A'\subset A$ and a Lipschitz onto map $f\colon A'\to B$.
As a corollary we prove that any `natural' dimension in $\mathbb{R}^n$ 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 $\mathbb{R}$. The quantity we introduce for this turns out to be closely related to the upper box dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02639 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lipschitz images and dimensions Balka, Richárd Keleti, Tamás Classical Analysis and ODEs Metric Geometry 28A78, 28A80, 51F30, 54E45 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 $A'\subset A$ and a Lipschitz onto map $f\colon A'\to B$. As a corollary we prove that any `natural' dimension in $\mathbb{R}^n$ 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 $\mathbb{R}$. The quantity we introduce for this turns out to be closely related to the upper box dimension. |
| title | Lipschitz images and dimensions |
| topic | Classical Analysis and ODEs Metric Geometry 28A78, 28A80, 51F30, 54E45 |
| url | https://arxiv.org/abs/2308.02639 |