Structures of moduli spaces of generalized Cantor sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908906358308864 |
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| author | Shiga, Hiroshige |
| author_facet | Shiga, Hiroshige |
| contents | For each $ω\in (0, 1)^{\mathbb N}$, we may construct a Cantor set $E(ω)\subset [0, 1]$ called a generalized Cantor set for $ω$. We study the moduli space of $ω$ denoted by $\mathcal M(ω)\subset (0, 1)^{\mathbb N}$. It is the set of $ω'$ so that $E(ω')$ is quasiconformally equivalent to $E(ω)$. In this paper, we show that the set $\mathcal M(ω)$ is measurable in $(0, 1)^{\mathbb N}$ and we give a necessary condition for $ω'$ to belong to $\mathcal M(ω)$. By using this condition, we show that there are uncountably many moduli spaces in $(0, 1)^{\mathbb N}$. We also show that except for at most one moduli space, the volume of the moduli space with respect to the standard product measure of $(0, 1)^{\mathbb N}$ vanishes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13990 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structures of moduli spaces of generalized Cantor sets Shiga, Hiroshige Complex Variables Geometric Topology 30C62 For each $ω\in (0, 1)^{\mathbb N}$, we may construct a Cantor set $E(ω)\subset [0, 1]$ called a generalized Cantor set for $ω$. We study the moduli space of $ω$ denoted by $\mathcal M(ω)\subset (0, 1)^{\mathbb N}$. It is the set of $ω'$ so that $E(ω')$ is quasiconformally equivalent to $E(ω)$. In this paper, we show that the set $\mathcal M(ω)$ is measurable in $(0, 1)^{\mathbb N}$ and we give a necessary condition for $ω'$ to belong to $\mathcal M(ω)$. By using this condition, we show that there are uncountably many moduli spaces in $(0, 1)^{\mathbb N}$. We also show that except for at most one moduli space, the volume of the moduli space with respect to the standard product measure of $(0, 1)^{\mathbb N}$ vanishes. |
| title | Structures of moduli spaces of generalized Cantor sets |
| topic | Complex Variables Geometric Topology 30C62 |
| url | https://arxiv.org/abs/2512.13990 |