Self-similar sets and Lipschitz graphs
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917125975703552 |
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| author | Davey, Blair Ghinassi, Silvia Wilson, Bobby |
| author_facet | Davey, Blair Ghinassi, Silvia Wilson, Bobby |
| contents | We investigate and quantify the distinction between rectifiable and purely unrectifiable 1-sets in the plane. That is, given that purely unrectifiable 1-sets always have null intersections with Lipschitz images, we ask whether these sets intersect with Lipschitz images at a dimension that is close to one. In an answer to this question, we show that one-dimensional attractors of iterated function systems that satisfy the open set condition have subsets of dimension arbitrarily close to one that can be covered by Lipschitz graphs. Moreover, the Lipschitz constant of such graphs depends explicitly on the difference between the dimension of the original set and the subset that intersects with the graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04837 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-similar sets and Lipschitz graphs Davey, Blair Ghinassi, Silvia Wilson, Bobby Classical Analysis and ODEs Metric Geometry 28A75, 28A78, 28A80 We investigate and quantify the distinction between rectifiable and purely unrectifiable 1-sets in the plane. That is, given that purely unrectifiable 1-sets always have null intersections with Lipschitz images, we ask whether these sets intersect with Lipschitz images at a dimension that is close to one. In an answer to this question, we show that one-dimensional attractors of iterated function systems that satisfy the open set condition have subsets of dimension arbitrarily close to one that can be covered by Lipschitz graphs. Moreover, the Lipschitz constant of such graphs depends explicitly on the difference between the dimension of the original set and the subset that intersects with the graph. |
| title | Self-similar sets and Lipschitz graphs |
| topic | Classical Analysis and ODEs Metric Geometry 28A75, 28A78, 28A80 |
| url | https://arxiv.org/abs/2407.04837 |