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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
##### Keywords:
quasiconformal mappings; quasicircles; Hausdorff measure
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##### References:
 [1] Astala, K., Area distortion of quasiconformal mappings. Acta Math., 173 (1994), 37–60. · Zbl 0815.30015 [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
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