Quasicircle
In geometry, a quasicircle is a Jordan curve which is the image of a Euclidean circle under a quasiconformal mapping of the plane to itself.
A quasicircle may also be characterised as a Jordan curve which is set-wise invariant under a quasiconformal mapping.[1] Lars Ahlfors showed that a finite Jordan curve is a quasicircle if and only if the cross-ratio of any four points is bounded.[2]
Quasiconformal mappings do not necessarily preserve Hausdorff dimension. It has been established by Stanislav Smirnov[3] 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
References
- ^ Donald K. Blevins; Bruce P. Palka (July 1975). "A characterization of quasicircles". Proceedings of the American Mathematical Society. 50 (1): 328–331.
- ^ Lars V. Ahlfors (1963). "Quasiconformal reflections". Acta Mathematica. 109: 291–301. Zbl 0121.06403.
- ^ Stanislav Smirnov (2010). "Dimension of quasicircles". Acta Mathematica. 205: 189–197. doi:10.1007/s11511-010-0053-8. MR 2011j:30027.
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