Quasiconformal characterization of Schottky sets
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914526740348928 |
|---|---|
| author | Ntalampekos, Dimitrios |
| author_facet | Ntalampekos, Dimitrios |
| contents | The complement of the union of a collection of disjoint open disks in the $2$-sphere is called a Schottky set. We prove that a subset $S$ of the $2$-sphere is quasiconformally equivalent to a Schottky set if and only if every pair of complementary components of $S$ can be mapped to a pair of open disks with a uniformly quasiconformal homeomorphism of the sphere. Our theorem applies to Sierpiński carpets and gaskets, yielding for the first time a general quasiconformal uniformization result for gaskets. Moreover, it contains Bonk's uniformization result for carpets as a special case and does not rely on the condition of uniform relative separation that is used in relevant works. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22658 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasiconformal characterization of Schottky sets Ntalampekos, Dimitrios Complex Variables Dynamical Systems Metric Geometry Primary 30C62, 30C20, Secondary 30F10, 37F32 The complement of the union of a collection of disjoint open disks in the $2$-sphere is called a Schottky set. We prove that a subset $S$ of the $2$-sphere is quasiconformally equivalent to a Schottky set if and only if every pair of complementary components of $S$ can be mapped to a pair of open disks with a uniformly quasiconformal homeomorphism of the sphere. Our theorem applies to Sierpiński carpets and gaskets, yielding for the first time a general quasiconformal uniformization result for gaskets. Moreover, it contains Bonk's uniformization result for carpets as a special case and does not rely on the condition of uniform relative separation that is used in relevant works. |
| title | Quasiconformal characterization of Schottky sets |
| topic | Complex Variables Dynamical Systems Metric Geometry Primary 30C62, 30C20, Secondary 30F10, 37F32 |
| url | https://arxiv.org/abs/2507.22658 |