quasicircle

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.{{harvnb|Lehto|1987|p=38}}

As shown in {{harvtxt|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.{{harvnb|Lehto|Virtanen|1973|pp=97–98}}

{{harvtxt|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{{harvnb|Carleson|Gamelin|1993|p=102}}

:$|z\_1-z\_3|\; +\; |z\_2-z\_3|\; \backslash le\; C\; |z\_1-z\_2|.$

This property is also called bounded turning{{harvnb|Lehto|Virtanen|1973|pp=100–102}} or the arc condition.{{harvnb|Krzyz|1983|p=45}}

For Jordan curves in the extended plane passing through ∞, {{harvtxt|Ahlfors|1966}} gave a simpler necessary and sufficient condition to be a quasicircle.{{harvnb|Ahlfors|1966|p=81}}{{harvnb|Krzyz|1983|pp=48–49}} 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

:$\backslash displaystyle\{\backslash left|z\_3\; -\{z\_1+z\_2\backslash over\; 2\}\backslash right|\backslash le\; C\; |z\_1-z\_2|.\}$

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

:$C\_1|s-t|\backslash le\; |f(s)-f(t)|\; \backslash le\; C\_2\; |s-t|$

for positive constants C_i.{{harvnb|Lehto|Virtanen|1973|pp=104–105}}

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

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

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 correspondence shows that the space of quasicircles can also be taken as a model.{{harvnb|Krzyz|1983|p={{pn|date=September 2023}}}}

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

:$\backslash displaystyle\{R\_0(z)\; =\; \{1\backslash over\; \backslash overline\{z\}\}\}$

provides such a reflection for the unit circle, any quasicircle admits a quasiconformal reflection. {{harvtxt|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.{{harvnb|Pommerenke|1975|pp=271–281}} If f extends to a quasiconformal mapping of the extended complex plane then ∂Ω is by definition a quasicircle. Conversely {{harvtxt|Ahlfors|1963}} observed that if ∂Ω is a quasicircle and R₁ denotes the quasiconformal reflection in ∂Ω then the assignment

:$\backslash displaystyle\{f(z)=R\_1f\; R\_0(z)\}$

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

File:Flocke.PNG]]

Quasicircles were known to arise as the Julia sets of rational maps R(z). {{harvtxt|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

:$|\backslash partial\_z\; R^n(z)|\; \backslash ge\; c\; A^n$

on the Julia set, then the Julia set is a quasicircle.

There are many examples:{{harvnb|Carleson|Gamelin|1993|pp=123–126}}{{harvnb|Rohde|1991}}

• 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 are obtained as quasiconformal deformations of Fuchsian groups. By definition their limit sets are quasicircles.{{harvnb|Bers|1961}}{{harvnb|Bowen|1979}}{{harvnb|Mumford|Series|Wright|2002}}{{harvnb|Imayoshi|Taniguchi|1992|p=147}}{{harvnb|Marden|2007|pp=79–80,134}}

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.

:$\backslash mu(g(z))\{\backslash overline\{\backslash partial\_\{z\}g(z)\}\backslash over\; \backslash partial\_z\; g(z)\}=\backslash mu(z)$

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

:$\backslash partial\_\{\backslash overline\{z\}\}\; f\; (z)\; =\backslash mu(z)\backslash partial\_zf(z)$

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

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

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.

File:Douady rabbit.png is composed of quasicircles with Hausdorff dimension approximately 1.3934{{harvnb|Carleson|Gamelin|1993|p=122}} ]]

It is known that there are quasicircles for which no segment has finite length.{{harvnb|Lehto|Virtanen|1973|p=104}} The Hausdorff dimension of quasicircles was first investigated by {{harvtxt|Gehring|Väisälä|1973}}, who proved that it can take all values in the interval [1,2).{{harvnb|Lehto|1987|p=38}} {{harvtxt|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{{harvnb|Astala|Iwaniec|Martin|2009}}

:$d\_H(C)\; \backslash le\; 1\; +\; k$

where

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

On the other hand, the Hausdorff dimension 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

:

Since these are quasicircles corresponding to a dilatation

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

where

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

this led {{harvtxt|Becker|Pommerenke|1987}} to show that for k small

:$1+\; 0.36\; k^2\backslash le\; d\_H(C)\; \backslash le\; 1\; +\; 37\; k^2.$

Having improved the lower bound following calculations for the Koch snowflake with Steffen Rohde and Oded Schramm,

{{harvtxt|Astala|1994}} conjectured that

:$d\_H(C)\; \backslash le\; 1\; +\; k^2.$

This conjecture was proved by {{harvtxt|Smirnov|2010}}; a complete account of his proof, prior to publication, was already given in {{harvtxt|Astala|Iwaniec|Martin|2009}}.

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

:$\backslash lambda=d(2-d)\backslash ,\backslash in\; (0,1)$

is the lowest eigenvalue of the Laplacian of the corresponding hyperbolic 3-manifold.{{harvnb|Astala|Zinsmeister|1994}}{{harvnb|Marden|2007|p=284}}

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