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Dimension of quasicircles. (English) Zbl 1211.30037
In his paper on the area and Hausdorff dimension distortion of plane quasiconformal mappings [Acta Math. 173, No. 1, 37–60 (1994; Zbl 0815.30015)], K. Astala made the conjecture that the Hausdorff dimension of a \(k\)-quasicircle cannot exceed \(1+k^2\). A \(k\)-quasicircle is the image of a circle under a quasiconformal map of the plane whose complex dilatation \(\mu\) satisfies \(|\mu| \leq k\) a.e. From his theory it follows that the Hausdorff dimension of a \(k\)-quasicircle is always \(\leq 1+ k\), but this does not take into account the special character of a quasicircle. In fact it was known, see [J. Becker and Ch. Pommerenke, Ann. Acad. Sci. Fenn., Ser. A I, Math. 12, No. 2, 329–333 (1987; Zbl 0606.30023)], that for small values of \(k\), the Hausdorff dimension of a \(k\)-quasicircle is strictly less than \(1+k\).
In this paper, the author proves the Astala conjecture. In the proof, the special character of a quasicircle is taken into account by representing quasicircles by antisymmetric Beltrami coefficients; in fact, for \(k\)-quasilines, a special representation theorem makes use of this antisymmetry. The proof of the Astala conjecture then relies on this and a symmetric Harnack inequality. Holomorphic motions and the thermodynamic formalism are also employed. It is not known if the upper bound \(1+k^2\) is sharp.

MSC:
30C62 Quasiconformal mappings in the complex plane
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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References:
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[2] Becker, J. & Pommerenke, C., On the Hausdorff dimension of quasicircles. Ann. Acad. Sci. Fenn. Ser. A I Math., 12 (1987), 329–333. · Zbl 0606.30023
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