Dimension of Quasicircles

We introduce canonical antisymmetric quasiconformal maps, which minimize the quasiconformality constant among maps sending the unit circle to a given quasicircle. As an application we prove Astala’s conjecture that the Hausdorff dimension of a k-quasicircle is at most 1 + k. A homeomorphism φ of planar domains is called k-quasiconformal, if it belongs locally to the Sobolev class W 1 2 and its Beltrami coefficient μφ(z) := ∂̄φ(z)/∂φ(z), has bounded L∞ norm: ‖φ‖ ≤ k < 1. Equivalently one can demand that for almost every point z in the domain of definition directional derivatives satisfy max α |∂αf(z)| ≤ Kmin α |∂αf(z)| , where the constants of quasiconformality are related by k = K − 1 K + 1 ∈ [0, 1[, K = 1 + k 1− k ∈ [1,∞[. Quasiconformal maps change eccentricities of infinitesimal ellipses at most by a factor of K, and it is common to visualize a quasiconformal map φ by considering a measurable field M(z) of infinitesimal ellipses, which is mapped by φ to the field of infinitesimal circles. More rigorously, M(z) is an ellipse on the tangent bundle, centered at 0 and defined up to homothety. Being preimage of a circle under the differential of φ at z, in the complex coordinate v ∈ TzC it is given by the equation |v + v̄μφ(z)| = const, and so its eccentricity is equal to |1 + |μφ(z)|| / |1− |μφ(z)|| ≤ K. Quasiconformal maps constitute an important generalization of conformal maps, which one obtains for K = 1 (or k = 0). However, while conformal maps preserve the Hausdorff dimension, quasiconformal maps can alter it. Understanding this phenomenon was a major challenge until the work [1] of Astala, where he obtained sharp estimates for the area and dimension distortion in terms of the quasiconformality constants. Particularly, the image of a set of Hausdorff dimension 1 under k-quasiconformal map can have dimension at most 1 + k, which can be attained for certain Cantor-type sets (like the Garnett-Ivanov square fractal). Nevertheless this work left open the question of the dimensional distortion of the subsets of smooth curves. Astala’s [1] implies that a k-quasicircle (that is an image of a circle under 1991 Mathematics Subject Classification. 30C62; 30C80. 1