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Hausdorff dimension of quasi-cirles of polygonal mappings and its applications

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Abstract

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 [µ] be a point in the universal Teichmüller space such that the Hausdorff dimension of f µ(ϖΔ) is bigger than one. We show that for every k n ∈ (0, 1) and polygonal differentials φ n , n = 1, 2, …, the sequence \( \{ [k_n \frac{{\overline {\phi _n } }} {{|\phi _n |}}]\} \) cannot converge to [µ] under the Teichmüller metric.

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Authors and Affiliations

  1. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, China

    ShengJin Huo & ShengJian Wu

  2. School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, 550001, China

    ShuAn Tang

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  1. ShengJin Huo
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  2. ShuAn Tang
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  3. ShengJian Wu
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Correspondence to ShuAn Tang.

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Huo, S., Tang, S. & Wu, S. Hausdorff dimension of quasi-cirles of polygonal mappings and its applications. Sci. China Math. 56, 1033–1040 (2013). https://doi.org/10.1007/s11425-012-4458-z

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  • DOI: https://doi.org/10.1007/s11425-012-4458-z

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