Quasicircle

In mathematics, a quasicircle, or quasiconformal curve, is a Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself. 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 and conformal welding. Quasicircles also play an important role in complex dynamical systems.

Geometric characterization

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. He also proved that they can be be characterized by in terms of a reverse trinagle inequality for three points: if two points A and B are chosen on the curve and C lies on the shorter of the resulting arcs, then the quantities (|AC| + |BC|)/|AB| should be bounded above.

Dynamical systems

Quasiconformal mappings do not necessarily preserve Hausdorff dimension. Smirnov (2010) proved that a quasicircle which is the image of a circle under a K-quasiconformal map has Hausdorff dimension bounded above by 1 + k² where the parameter k is defined as

$k={\frac {K-1}{K+1}}.$

References

• Ahlfors, Lars V. (1966), Lectures on quasiconformal mappings, Van Nostrand
• Ahlfors, L. (1963), "Quasiconformal reflections", Acta Mathematica, 109: 291–301, Zbl 0121.06403
• Carleson, L.; Gamelin, T. D. W. (1993), Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, ISBN 0-387-97942-5
• Imayoshi, Y.; Taniguchi, M. (1992), An Introduction to Teichmüller spaces, Springer-Verlag, ISBN 0-387-70088-9 +
• Lehto, O. (1987), Univalent functions and Teichmüller spaces, Springer-Verlag, pp. 50–59, 111–118, 196–205, ISBN 0387963103
• Lehto, O.; Virtanen, K. I. (1973), Quasiconformal mappings in the plane, Die Grundlehren der mathematischen Wissenschaften, vol. 126 (Second ed.), Springer-Verlag
• Smirnov, S. (2010), "Dimension of quasicircles", Acta Mathematica, 205: 189–197, doi:10.1007/s11511-010-0053-8, MR 2011j:30027 {{citation}}: Check |mr= value (help)