Smirnov, Stanislav Dimension of quasicircles. (English) Zbl 1211.30037 Acta Math. 205, No. 1, 189-197 (2010). 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. Reviewer: Olli Martio (Helsinki) Cited in 4 ReviewsCited in 18 Documents MSC: 30C62 Quasiconformal mappings in the complex plane 30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination Keywords:quasiconformal mappings; quasicircles; Hausdorff measure Citations:Zbl 0815.30015; Zbl 0606.30023 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Astala, K., Area distortion of quasiconformal mappings. Acta Math., 173 (1994), 37–60. · Zbl 0815.30015 · doi:10.1007/BF02392568 [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 [3] Prause, I., A remark on quasiconformal dimension distortion on the line. Ann. Acad. Sci. Fenn. Math., 32 (2007), 341–352. · Zbl 1122.30014 [4] Prause, I. & Smirnov, S., Quasisymmetric distortion spectrum. Preprint, 2009. arXiv:0910.4723v1 [math.CV]. · Zbl 1235.30017 [5] Ruelle, D., Thermodynamic Formalism. Encyclopedia of Mathematics and its Applications, 5. Addison-Wesley, Reading, MA, 1978. · Zbl 0401.28016 [6] – Repellers for real analytic maps. Ergodic Theory Dynamical Systems, 2 (1982), 99–107. · Zbl 0506.58024 [7] Väläisä, J., Lectures on n-Dimensional Quasiconformal Mappings. Lecture Notes in Mathematics, 229. Springer, Berlin–Heidelberg, 1971. · Zbl 0221.30031 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.