Stretching and Rotation Sets of Quasiconformal Mappings
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2017
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| _version_ | 1866908048694444032 |
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| author | Bongers, Rosemarie |
| author_facet | Bongers, Rosemarie |
| contents | Quasiconformal maps in the plane are orientation preserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of bounded eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible and powerful generalization of conformal mappings. In this work, we study the singularities of these maps, in particular the sizes of the sets where a quasiconformal map can exhibit given stretching and rotation behavior. We improve results by Astala-Iwaniec-Prause-Saksman and Hitruhin to give examples of stretching and rotation sets with non-sigma-finite measure at the appropriate Hausdorff dimension. We also improve this to give examples with positive Riesz capacity at the critical homogeneity, as well as positivity for a broad class of gauged Hausdorff measures at that dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1710_04341 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Stretching and Rotation Sets of Quasiconformal Mappings Bongers, Rosemarie Complex Variables 30C62, 28A78 Quasiconformal maps in the plane are orientation preserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of bounded eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible and powerful generalization of conformal mappings. In this work, we study the singularities of these maps, in particular the sizes of the sets where a quasiconformal map can exhibit given stretching and rotation behavior. We improve results by Astala-Iwaniec-Prause-Saksman and Hitruhin to give examples of stretching and rotation sets with non-sigma-finite measure at the appropriate Hausdorff dimension. We also improve this to give examples with positive Riesz capacity at the critical homogeneity, as well as positivity for a broad class of gauged Hausdorff measures at that dimension. |
| title | Stretching and Rotation Sets of Quasiconformal Mappings |
| topic | Complex Variables 30C62, 28A78 |
| url | https://arxiv.org/abs/1710.04341 |