Wikipedia

Quasicircle

Not to be confused with Pseudocircle.

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

Definitions edit

A quasicircle is defined as the image of a circle under a quasiconformal mapping of the extended complex plane. 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. In particular a quasicircle is a Jordan curve. The interior of a quasicircle is called a quasidisk.[3]

As shown in Lehto & Virtanen (1973), where the older term "quasiconformal curve" is used, if a Jordan curve is the image of a circle under a quasiconformal map in a neighbourhood of the curve, then it is also the image of a circle under a quasiconformal mapping of the extended plane and thus a quasicircle. The same is true for "quasiconformal arcs" which can be defined as quasiconformal images of a circular arc either in an open set or equivalently in the extended plane.[4]

Geometric characterizations edit

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.

Ahlfors also proved that quasicircles can be characterized in terms of a reverse triangle inequality for three points: there should be a constant C such that if two points z1 and z2 are chosen on the curve and z3 lies on the shorter of the resulting arcs, then[5]

 

This property is also called bounded turning[6] or the arc condition.[7]

For Jordan curves in the extended plane passing through ∞, Ahlfors (1966) gave a simpler necessary and sufficient condition to be a quasicircle.[8][9] There is a constant C > 0 such that if z1, z2 are any points on the curve and z3 lies on the segment between them, then

 

These metric characterizations imply that an arc or closed curve is quasiconformal whenever it arises as the image of an interval or the circle under a bi-Lipschitz map f, i.e. satisfying

 

for positive constants Ci.[10]

Quasicircles and quasisymmetric homeomorphisms edit

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. The maps f and g extend continuously to the circle |z| = 1 and the sewing equation

 

holds. The image of the circle is a quasicircle.

Conversely, using the Riemann mapping theorem, the conformal maps f and g uniformizing the outside of a quasicircle give rise to a quasisymmetric homeomorphism through the above equation.

The quotient space of the group of quasisymmetric homeomorphisms by the subgroup of Möbius transformations provides a model of universal Teichmüller space. The above correspondence shows that the space of quasicircles can also be taken as a model.[11]

Quasiconformal reflection edit

A quasiconformal reflection in a Jordan curve is an orientation-reversing quasiconformal map of period 2 which switches the inside and the outside of the curve fixing points on the curve. Since the map

 

provides such a reflection for the unit circle, any quasicircle admits a quasiconformal reflection. Ahlfors (1963) proved that this property characterizes quasicircles.

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). As Carathéodory had proved using his theory of prime ends, f extends continuously to the unit circle if and only if ∂Ω is locally connected, i.e. admits a covering by finitely many compact connected sets of arbitrarily small diameter. The extension to the circle is 1-1 if and only if ∂Ω has no cut points, i.e. points which when removed from ∂Ω yield a disconnected set. 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.[12] If f extends to a quasiconformal mapping of the extended complex plane then ∂Ω is by definition a quasicircle. Conversely Ahlfors (1963) observed that if ∂Ω is a quasicircle and R1 denotes the quasiconformal reflection in ∂Ω then the assignment

 

for |z| > 1 defines a quasiconformal extension of f to the extended complex plane.

Complex dynamical systems edit

 
Koch snowflake

Quasicircles were known to arise as the Julia sets of rational maps R(z). Sullivan (1985) proved that if the Fatou set of R has two components and the action of R on the Julia set is "hyperbolic", i.e. there are constants c > 0 and A > 1 such that

 

on the Julia set, then the Julia set is a quasicircle.[5]

There are many examples:[13][14]

  • quadratic polynomials R(z) = z2 + c with an attracting fixed point
  • the Douady rabbit (c = –0.122561 + 0.744862i, where c3 + 2 c2 + c + 1 = 0)
  • quadratic polynomials z2 + λz with |λ| < 1
  • the Koch snowflake

Quasi-Fuchsian groups edit

Quasi-Fuchsian groups are obtained as quasiconformal deformations of Fuchsian groups. By definition their limit sets are quasicircles.[15][16][17][18][19]

Let Γ be a Fuchsian group of the first kind: a discrete subgroup of the Möbius group preserving the unit circle. acting properly discontinuously on the unit disk D and with limit set the unit circle.

Let μ(z) be a measurable function on D with

 

such that μ is Γ-invariant, i.e.

 

for every g in Γ. (μ is thus a "Beltrami differential" on the Riemann surface D / Γ.)

Extend μ to a function on C by setting μ(z) = 0 off D.

The Beltrami equation

 

admits a solution unique up to composition with a Möbius transformation.

It is a quasiconformal homeomorphism of the extended complex plane.

If g is an element of Γ, then f(g(z)) gives another solution of the Beltrami equation, so that

 

is a Möbius transformation.

The group α(Γ) is a quasi-Fuchsian group with limit set the quasicircle given by the image of the unit circle under f.

Hausdorff dimension edit

 
The Douady rabbit is composed of quasicircles with Hausdorff dimension approximately 1.3934[20]

It is known that there are quasicircles for which no segment has finite length.[21] The Hausdorff dimension of quasicircles was first investigated by Gehring & Väisälä (1973), who proved that it can take all values in the interval [1,2).[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).

For a quasi-Fuchsian group Bowen (1979) and Sullivan (1982) showed that the Hausdorff dimension d of the limit set is always greater than 1. When d < 2, the quantity

 

is the lowest eigenvalue of the Laplacian of the corresponding hyperbolic 3-manifold.[24][25]

Notes edit

  1. ^ Lehto & Virtanen 1973
  2. ^ Lehto 1983, p. 49.[full citation needed]
  3. ^ Lehto 1987, p. 38
  4. ^ Lehto & Virtanen 1973, pp. 97–98
  5. ^ a b Carleson & Gamelin 1993, p. 102
  6. ^ Lehto & Virtanen 1973, pp. 100–102
  7. ^ Lehto 1983, p. 45.[full citation needed]
  8. ^ Ahlfors 1966, p. 81
  9. ^ Lehto 1983, pp. 48–49.[full citation needed]
  10. ^ Lehto & Virtanen 1973, pp. 104–105
  11. ^ Lehto 1983, p. [page needed].[full citation needed]
  12. ^ Pommerenke 1975, pp. 271–281
  13. ^ Carleson & Gamelin 1993, pp. 123–126
  14. ^ Rohde 1991
  15. ^ Bers 1961
  16. ^ Bowen 1979
  17. ^ Mumford, Series & Wright 2002
  18. ^ Imayoshi & Taniguchi 1992, p. 147
  19. ^ Marden 2007, pp. 79–80, 134
  20. ^ Carleson & Gamelin 1993, p. 122
  21. ^ Lehto & Virtanen 1973, p. 104
  22. ^ Lehto 1982, p. 38.[full citation needed]
  23. ^ Astala, Iwaniec & Martin 2009
  24. ^ Astala & Zinsmeister 1994
  25. ^ Marden 2007, p. 284

References edit

  • Ahlfors, Lars V. (1966), Lectures on quasiconformal mappings, Van Nostrand
  • Ahlfors, L. (1963), "Quasiconformal reflections", Acta Mathematica, 109: 291–301, doi:10.1007/bf02391816, Zbl 0121.06403
  • Astala, K. (1993), "Distortion of area and dimension under quasiconformal mappings in the plane", Proc. Natl. Acad. Sci. U.S.A., 90 (24): 11958–11959, Bibcode:1993PNAS...9011958A, doi:10.1073/pnas.90.24.11958, PMC 48104, PMID 11607447
  • Astala, K.; Zinsmeister, M. (1994), "Holomorphic families of quasi-Fuchsian groups", Ergodic Theory Dynam. Systems, 14 (2): 207–212, doi:10.1017/s0143385700007847, S2CID 121209816
  • Astala, K. (1994), "Area distortion of quasiconformal mappings", Acta Math., 173: 37–60, doi:10.1007/bf02392568
  • Astala, Kari; Iwaniec, Tadeusz; Martin, Gaven (2009), Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton mathematical series, vol. 48, Princeton University Press, pp. 332–342, ISBN 978-0-691-13777-3, Section 13.2, Dimension of quasicircles.
  • Becker, J.; Pommerenke, C. (1987), "On the Hausdorff dimension of quasicircles", Ann. Acad. Sci. Fenn. Ser. A I Math., 12: 329–333, doi:10.5186/aasfm.1987.1206
  • Bers, Lipman (August 1961). "Uniformization by Beltrami equations". Communications on Pure and Applied Mathematics. 14 (3): 215–228. doi:10.1002/cpa.3160140304.
  • Bowen, R. (1979), "Hausdorff dimension of quasicircles", Inst. Hautes Études Sci. Publ. Math., 50: 11–25, doi:10.1007/BF02684767, S2CID 55631433
  • Carleson, L.; Gamelin, T. D. W. (1993), Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, ISBN 978-0-387-97942-7
  • Gehring, F. W.; Väisälä, J. (1973), "Hausdorff dimension and quasiconformal mappings", Journal of the London Mathematical Society, 6 (3): 504–512, CiteSeerX 10.1.1.125.2374, doi:10.1112/jlms/s2-6.3.504
  • Gehring, F. W. (1982), Characteristic properties of quasidisks, Séminaire de Mathématiques Supérieures, vol. 84, Presses de l'Université de Montréal, ISBN 978-2-7606-0601-2
  • Imayoshi, Y.; Taniguchi, M. (1992), An Introduction to Teichmüller spaces, Springer-Verlag, ISBN 978-0-387-70088-5 +
  • Lehto, O. (1987), Univalent functions and Teichmüller spaces, Springer-Verlag, pp. 50–59, 111–118, 196–205, ISBN 978-0-387-96310-5
  • Lehto, O.; Virtanen, K. I. (1973), Quasiconformal mappings in the plane, Die Grundlehren der mathematischen Wissenschaften, vol. 126 (Second ed.), Springer-Verlag
  • Marden, A. (2007), Outer circles. An introduction to hyperbolic 3-manifolds, Cambridge University Press, ISBN 978-0-521-83974-7
  • Mumford, D.; Series, C.; Wright, David (2002), Indra's pearls. The vision of Felix Klein, Cambridge University Press, ISBN 978-0-521-35253-6
  • Pfluger, A. (1961), "Ueber die Konstruktion Riemannscher Flächen durch Verheftung", J. Indian Math. Soc., 24: 401–412
  • Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht
  • Rohde, S. (1991), "On conformal welding and quasicircles", Michigan Math. J., 38: 111–116, doi:10.1307/mmj/1029004266
  • Sullivan, D. (1982), "Discrete conformal groups and measurable dynamics", Bull. Amer. Math. Soc., 6: 57–73, doi:10.1090/s0273-0979-1982-14966-7
  • Sullivan, D. (1985), "Quasiconformal homeomorphisms and dynamics, I, Solution of the Fatou-Julia problem on wandering domains", Annals of Mathematics, 122 (2): 401–418, doi:10.2307/1971308, JSTOR 1971308
  • Tienari, M. (1962), "Fortsetzung einer quasikonformen Abbildung über einen Jordanbogen", Ann. Acad. Sci. Fenn. Ser. A, 321
  • Smirnov, S. (2010), "Dimension of quasicircles", Acta Mathematica, 205: 189–197, arXiv:0904.1237, doi:10.1007/s11511-010-0053-8, MR 2736155, S2CID 17945998
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