From Lipschitz embedding to Lipschitz equivalence between dust-like self-similar sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912575359287296 |
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| author | Ruan, Huo-Jun Xiao, Jian-Ci |
| author_facet | Ruan, Huo-Jun Xiao, Jian-Ci |
| contents | Let $K,F\subset\mathbb{R}^d$ be two dust-like self-similar sets sharing the same Hausdorff dimension. We consider when the mere existence of a Lipschitz embedding from $K$ to $F$ already implies their Lipschitz equivalence. Our main result is threefold: (1) if the Lipschitz image of $K$ intersects $F$ in a set of positive Hausdorff measure, then $K$ admits a Lipschitz surjection onto $F$; (2) if $F$ is in addition homogeneous, then the generating iterated function systems of $K, F$ should have algebraically dependent ratios and consequently, $K$ and $F$ are Lipschitz equivalent; (3) the Lipschitz equivalence can fail without the homogeneity assumption. This answers two questions in Balka and Keleti [Adv. Math. 446 (2024), 109669]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_07670 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Lipschitz embedding to Lipschitz equivalence between dust-like self-similar sets Ruan, Huo-Jun Xiao, Jian-Ci Classical Analysis and ODEs Metric Geometry Primary 28A80, Secondary 28A78, 51F30 Let $K,F\subset\mathbb{R}^d$ be two dust-like self-similar sets sharing the same Hausdorff dimension. We consider when the mere existence of a Lipschitz embedding from $K$ to $F$ already implies their Lipschitz equivalence. Our main result is threefold: (1) if the Lipschitz image of $K$ intersects $F$ in a set of positive Hausdorff measure, then $K$ admits a Lipschitz surjection onto $F$; (2) if $F$ is in addition homogeneous, then the generating iterated function systems of $K, F$ should have algebraically dependent ratios and consequently, $K$ and $F$ are Lipschitz equivalent; (3) the Lipschitz equivalence can fail without the homogeneity assumption. This answers two questions in Balka and Keleti [Adv. Math. 446 (2024), 109669]. |
| title | From Lipschitz embedding to Lipschitz equivalence between dust-like self-similar sets |
| topic | Classical Analysis and ODEs Metric Geometry Primary 28A80, Secondary 28A78, 51F30 |
| url | https://arxiv.org/abs/2508.07670 |