Flexible curves and Hausdorff dimension
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912835264577536 |
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| author | Rodriguez, Alex |
| author_facet | Rodriguez, Alex |
| contents | We show that given a log-singular circle homeomorphism $h$ and given any $s\in[1,2]$, there is a flexible curve of Hausdorff dimension $s$ with welding $h$. We also see that there is another curve with welding $h$ and positive area. In particular, this implies that given a flexible curve $Γ$, there is a homeomorphism of the plane $ϕ\colon\mathbb{C}\to\mathbb{C}$, conformal off $Γ$, so that $ϕ(Γ)$ has positive area. This answers a particular case of the corresponding conjecture for general non-conformally removable sets, for a class of curves that is residual in the space of all Jordan curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14125 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Flexible curves and Hausdorff dimension Rodriguez, Alex Complex Variables Primary 30C85, Secondary 30E25 We show that given a log-singular circle homeomorphism $h$ and given any $s\in[1,2]$, there is a flexible curve of Hausdorff dimension $s$ with welding $h$. We also see that there is another curve with welding $h$ and positive area. In particular, this implies that given a flexible curve $Γ$, there is a homeomorphism of the plane $ϕ\colon\mathbb{C}\to\mathbb{C}$, conformal off $Γ$, so that $ϕ(Γ)$ has positive area. This answers a particular case of the corresponding conjecture for general non-conformally removable sets, for a class of curves that is residual in the space of all Jordan curves. |
| title | Flexible curves and Hausdorff dimension |
| topic | Complex Variables Primary 30C85, Secondary 30E25 |
| url | https://arxiv.org/abs/2601.14125 |