Graphs that are not minimal for conformal dimension
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912161906819072 |
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| author | Romney, Matthew |
| author_facet | Romney, Matthew |
| contents | We construct functions $f \colon [0,1] \to [0,1]$ whose graph as a subset of $\mathbb{R}^2$ has Hausdorff dimension greater than any given value $α\in (1,2)$ but conformal dimension $1$. These functions have the property that a positive proportion of level sets have positive codimension-$1$ measure. This result gives a negative answer to a question of Binder--Hakobyan--Li. We also give a function whose graph has Hausdorff dimension $2$ but conformal dimension $1$. The construction is based on the author's previous solution to the inverse absolute continuity problem for quasisymmetric mappings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15016 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Graphs that are not minimal for conformal dimension Romney, Matthew Metric Geometry 28A78, 30L10 We construct functions $f \colon [0,1] \to [0,1]$ whose graph as a subset of $\mathbb{R}^2$ has Hausdorff dimension greater than any given value $α\in (1,2)$ but conformal dimension $1$. These functions have the property that a positive proportion of level sets have positive codimension-$1$ measure. This result gives a negative answer to a question of Binder--Hakobyan--Li. We also give a function whose graph has Hausdorff dimension $2$ but conformal dimension $1$. The construction is based on the author's previous solution to the inverse absolute continuity problem for quasisymmetric mappings. |
| title | Graphs that are not minimal for conformal dimension |
| topic | Metric Geometry 28A78, 30L10 |
| url | https://arxiv.org/abs/2412.15016 |