New Hausdorff type dimensions and optimal bounds for bilipschitz invariant dimensions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918520359485440 |
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| author | Balka, Richárd Keleti, Tamás |
| author_facet | Balka, Richárd Keleti, Tamás |
| contents | We introduce a new family of fractal dimensions by restricting the set of diameters in the coverings in the usual definition of the Hausdorff dimension. Among others, we prove that this family contains continuum many distinct dimensions, and they share most of the properties of the Hausdorff dimension, which answers negatively a question of Fraser. On the other hand, we also prove that among these new dimensions only the Hausdorff dimension behaves nicely with respect to Hölder functions.
We also consider the supremum of these new dimensions, which turns out to be another interesting notion of fractal dimension.
We prove that among those bilipschitz invariant, monotone dimensions on the compact subsets of $\mathbb{R}^n$ that agree with the similarity dimension for the simplest self-similar sets, the modified lower dimension is the smallest and when $n=1$ the Assouad dimension is the greatest, and this latter statement is false for $n>1$. This answers a question of Rutar. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06456 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | New Hausdorff type dimensions and optimal bounds for bilipschitz invariant dimensions Balka, Richárd Keleti, Tamás Classical Analysis and ODEs Metric Geometry 28A78, 28A80, 51F30 We introduce a new family of fractal dimensions by restricting the set of diameters in the coverings in the usual definition of the Hausdorff dimension. Among others, we prove that this family contains continuum many distinct dimensions, and they share most of the properties of the Hausdorff dimension, which answers negatively a question of Fraser. On the other hand, we also prove that among these new dimensions only the Hausdorff dimension behaves nicely with respect to Hölder functions. We also consider the supremum of these new dimensions, which turns out to be another interesting notion of fractal dimension. We prove that among those bilipschitz invariant, monotone dimensions on the compact subsets of $\mathbb{R}^n$ that agree with the similarity dimension for the simplest self-similar sets, the modified lower dimension is the smallest and when $n=1$ the Assouad dimension is the greatest, and this latter statement is false for $n>1$. This answers a question of Rutar. |
| title | New Hausdorff type dimensions and optimal bounds for bilipschitz invariant dimensions |
| topic | Classical Analysis and ODEs Metric Geometry 28A78, 28A80, 51F30 |
| url | https://arxiv.org/abs/2312.06456 |