Hausdorff dimension of quasi-cirles of polygonal mappings and its applications. (English) Zbl 1269.30028
Summary: We show that the Hausdorff dimension of quasi-circles of polygonal mappings is one. Furthermore, we apply this result to the theory of extremal quasiconformal mappings. Let \([\mu ]\) be a point in the universal Teichmüller space such that the Hausdorff dimension of \(f _{\mu } (\partial; \Delta)\) is bigger than one. We show that for every \(k _{n } \in (0, 1)\) and polygonal differentials \(\varphi _{n }, n = 1, 2, \dots \), the sequence \( \{ [k_n \frac{{\overline {\varphi _n } }} {{|\varphi _n |}}]\} \) cannot converge to \([\mu ]\) under the Teichmüller metric.
MSC:
| 30C62 | Quasiconformal mappings in the complex plane |
| 30C75 | Extremal problems for conformal and quasiconformal mappings, other methods |
| 30F60 | Teichmüller theory for Riemann surfaces |
| 28A78 | Hausdorff and packing measures |
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