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

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 Teichmiiller space such that the Hausdorff dimension of fμ（δ△） is bigger than one. We show that for every kn ∈ （0, 1） and polygonal differentials δn, n = 1, 2, the sequence {[kn δn/｜δn｜} cannot converge to [μ] under the Teichmiiller metric.