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 Teichmüller space such that the Hausdorff dimension of f _µ(?Δ) 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.