Quasicircle

Originally introduced independently by Pfluger (1961) and Tienari (1962), in the older literature (in German) they were referred to as quasiconformal curves, a terminology which also applied to arcs.

[1][2] In complex analysis and geometric function theory, quasicircles play a fundamental role in the description of the universal Teichmüller space, through quasisymmetric homeomorphisms of the circle.

It is called a K-quasicircle if the quasiconformal mapping has dilatation K. The definition of quasicircle generalizes the characterization of a Jordan curve as the image of a circle under a homeomorphism of the plane.

[4] Ahlfors (1963) gave a geometric characterization of quasicircles as those Jordan curves for which the absolute value of the cross-ratio of any four points, taken in cyclic order, is bounded below by a positive constant.

[7] For Jordan curves in the extended plane passing through ∞, Ahlfors (1966) gave a simpler necessary and sufficient condition to be a quasicircle.

[10] If φ is a quasisymmetric homeomorphism of the circle, then there are conformal maps f of [z| < 1 and g of |z|>1 into disjoint regions such that the complement of the images of f and g is a Jordan curve.

Ahlfors noted that this result can be applied to uniformly bounded holomorphic univalent functions f(z) on the unit disk D. Let Ω = f(D).

Carathéodory's theorem shows that a locally set without cut points is just a Jordan curve and that in precisely this case is the extension of f to the closed unit disk a homeomorphism.

[22] Astala (1993), using the new technique of "holomorphic motions" was able to estimate the change in the Hausdorff dimension of any planar set under a quasiconformal map with dilatation K. For quasicircles C, there was a crude estimate for the Hausdorff dimension[23] where On the other hand, the Hausdorff dimension for the Julia sets Jc of the iterates of the rational maps had been estimated as result of the work of Rufus Bowen and David Ruelle, who showed that Since these are quasicircles corresponding to a dilatation where this led Becker & Pommerenke (1987) to show that for k small Having improved the lower bound following calculations for the Koch snowflake with Steffen Rohde and Oded Schramm, Astala (1994) conjectured that This conjecture was proved by Smirnov (2010); a complete account of his proof, prior to publication, was already given in Astala, Iwaniec & Martin (2009).