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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.

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)

### MSC:

30C62 | Quasiconformal mappings in the complex plane |

30C80 | Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination |

### 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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