Quasicircle

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. 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

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 defintion 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.

As shown in Lahto & 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 an circular either in an open set or equivalently in the extended plane.

Geometric characterizations

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 be characterized in terms of a reverse triangle 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 quantity (|AC| + |BC|)/|AB| should be bounded above. This property is also called bounded turning or the arc condition.

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

$\displaystyle {\left|z_{3}-{z_{1}+z_{2} \over 2}\right|\leq C|z_{1}-z_{2}|.}$

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

$C_{1}|s-t|\leq |f(s)-f(t)|\leq C_{2}|s-t|$

for positive constants C_i.

Quasicircles and quasisymmetric homeomorphisms

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

$\varphi =g^{-1}\circ f$

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 correpondence shows that the space of quasicircles can also be taken as a model.

Quasiconformal reflection

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

$j(z)={1 \over {\overline {z}}}$

provides such a reflection for the unit circle, any quasicircle admits a quasiconformal reflection. Ahlfors (1966) proved that this property characterizes quasicircles.

Complex dynamical systems

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, then the Julia set is a quasicircle.

There are many examples:

• 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

Bers & 61 gave a construction of quasi-Fuchsian groups, with limit sets that are quasicircles.

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

$\|\mu \|_{\infty }<1$

such that μ is Γ-invariant.

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

The Beltrami equation

$\partial _{\overline {z}}f(z)=\mu (z)\partial _{z}f(z)$

admits a unique solution fixing 0, 1 and ∞.

It is a quasiconformal homeomorphism of the extended complex plane.

If g is an element of Γ, then

$\alpha (g)=f\circ g\circ f^{-1}$

is holomorphic and so 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

It is known that there are quasicircles for which no segment has finite length. 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). 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

$d_{H}(C)\leq 1+k$

where

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

On the other hand, the Hausdorff dimension was known for the Julia sets J_c of the iterates of the rational maps

$R(z)=z^{2}+c$

had been estimated as result of the work of Rufus Bowen and David Ruelle, who showed that

$1<d_{H}(J_{c})<1+{|c|^{2} \over 4\log 2}+o(|c|^{2}).$

Since these are quasicircles corresponding to a dilatation

$K={\sqrt {1+t \over 1-t}}$

where

$t=|1-{\sqrt {1-4c}}|,$

this led Becker & Pommerenke (1987) to show that for k small

$1+0.36k^{2}\leq d_{H}(C)\leq 1+37k^{2}.$

Having improved the lower bound following calculations for the Koch snowflake with Steffen Rohde and Oded Schramm, Astala (1994) conjectured that

$d_{H}(C)\leq 1+k^{2}.$

This conjecture was proved by Smirnov (2010); a complete account of his proof, prior to publication, was already given in Astala, Iwaniec & Martin (2009).

References

• Ahlfors, Lars V. (1966), Lectures on quasiconformal mappings, Van Nostrand
• Ahlfors, L. (1963), "Quasiconformal reflections", Acta Mathematica, 109: 291–301, Zbl 0121.06403
• Astala, K. (1993), "Distortion of area and dimension under quasiconformal mappings in the plane", Proc. Nat. Acad. Sci. U.S.A., 90: 11958–11959
• Astala, K. (1994), "Area distortion of quasiconformal mappings", Acta Math., 173: 37–60
• 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 0691137773 {{citation}}: horizontal tab character in |id= at position 5 (help), 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
• Bowen, R. (1979), "Hausdorff dimension of quasicircles", Inst. Hautes Études Sci. Publ. Math., 50: 11–25
• Carleson, L.; Gamelin, T. D. W. (1993), Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, ISBN 0-387-97942-5
• Gehring, F. W.; Väisälä, J. (1973), "Hausdorff dimension and quasiconformal mappings", J. London Math. Soc., 6: 504–512
• Gehring, F. W. (1982), Characteristic properties of quasidisks, Séminaire de Mathématiques Supérieures, vol. 84, Presses de l'Université de Montréal, ISBN 2-7606-0601-5
• 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
• Mumford, D.; Series, C.; Wright, David (2002), Indra's pearls. The vision of Felix Klein, Cambridge University Press, ISBN 0-521-35253-3
• Pfluger, A. (1961), "Ueber die Konstruktion Riemannscher Flächen durch Verheftung", J. Indian Math. Soc., 24 {{citation}}: Text "pages401–412" ignored (help)
• Rohde, S. (1991), "On conformal welding and quasicircles", Michigan Math. J., 38: 111–116
• Sullivan, D. (1985), "Quasiconformal homeomorphisms and dynamics, I, Solution of the Fatou-Julia problem on wandering domains", Annals of Math., 122: 401–418
• 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, doi:10.1007/s11511-010-0053-8, MR 2011j:30027 {{citation}}: Check |mr= value (help)