Quasicircle

{{Short description|Quasiconformal complex image of a circle}} {{distinguish|Pseudocircle}} {{CS1 config|mode=cs2}} 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 {{harvtxt|Pfluger|1961}} and {{harvtxt|Tienari|1962}}, in the older literature (in German) they were referred to as '''quasiconformal curves''', a terminology which also applied to arcs.<ref>{{harvnb|Lehto|Virtanen|1973}}</ref><ref>{{harvnb|Krzyz|1983|p=49}}</ref> 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 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''.<ref>{{harvnb|Lehto|1987|p=38}}</ref>

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.<ref>{{harvnb|Lehto|Virtanen|1973|pp=97–98}}</ref>

==Geometric characterizations== {{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 ''z''₁ and ''z''₂ are chosen on the curve and ''z''₃ lies on the shorter of the resulting arcs, then<ref name=autogenerated1>{{harvnb|Carleson|Gamelin|1993|p=102}}</ref>

:<math> |z_1-z_3| + |z_2-z_3| \le C |z_1-z_2|.</math>

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

For Jordan curves in the extended plane passing through ∞, {{harvtxt|Ahlfors|1966}} gave a simpler necessary and sufficient condition to be a quasicircle.<ref>{{harvnb|Ahlfors|1966|p=81}}</ref><ref>{{harvnb|Krzyz|1983|pp=48–49}}</ref> There is a constant ''C'' > 0 such that if ''z''₁, ''z''₂ are any points on the curve and ''z''₃ lies on the segment between them, then

:<math>\displaystyle{\left|z_3 -{z_1+z_2\over 2}\right|\le C |z_1-z_2|.}</math>

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

:<math> C_1|s-t|\le |f(s)-f(t)| \le C_2 |s-t|</math>

for positive constants ''C''_''i''.<ref>{{harvnb|Lehto|Virtanen|1973|pp=104–105}}</ref>

==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

:<math> \varphi= g^{-1}\circ f </math>

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.<ref>{{harvnb|Krzyz|1983|p={{pn|date=September 2023}}}}</ref>

==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

:<math>\displaystyle{R_0(z) = {1\over \overline{z}}} </math>

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.<ref>{{harvnb|Pommerenke|1975|pp=271–281}}</ref> 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 ''R''₁ denotes the quasiconformal reflection in ∂Ω then the assignment

:<math> \displaystyle{f(z)=R_1f R_0(z)}</math>

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

==Complex dynamical systems== [[File:Flocke.PNG|thumb|Koch snowflake]] 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

:<math> |\partial_z R^n(z)| \ge c A^n </math>

on the Julia set, then the Julia set is a quasicircle.<ref name=autogenerated1 />

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

* quadratic polynomials ''R''(''z'') = ''z''² + ''c'' with an attracting fixed point * the Douady rabbit (''c'' = –0.122561 + 0.744862i, where ''c''³ + 2 ''c''² + ''c'' + 1 = 0) * quadratic polynomials ''z''² + λ''z'' with |λ| < 1 * the Koch snowflake

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

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

:<math>\|\mu\|_\infty < 1</math>

such that μ is Γ-invariant, i.e.

:<math>\mu(g(z)){\overline{\partial_{z}g(z)}\over \partial_z g(z)}=\mu(z)</math>

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

:<math> \partial_{\overline{z}} f (z) =\mu(z)\partial_zf(z)</math>

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

:<math>\alpha(g)=f\circ g \circ f^{-1}</math>

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== [[File:Douady rabbit.png|thumb|The Douady rabbit is composed of quasicircles with Hausdorff dimension approximately 1.3934<ref>{{harvnb|Carleson|Gamelin|1993|p=122}}</ref> ]] It is known that there are quasicircles for which no segment has finite length.<ref>{{harvnb|Lehto|Virtanen|1973|p=104}}</ref> 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).<ref>{{harvnb|Lehto|1987|p=38}}</ref> {{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<ref>{{harvnb|Astala|Iwaniec|Martin|2009}}</ref>

:<math> d_H(C) \le 1 + k</math>

where

:<math>k={K-1\over K+1}.</math>

On the other hand, the Hausdorff dimension for the Julia sets ''J''_c of the iterates of the rational maps

:<math>R(z) =z^2 +c</math>

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

:<math>1 < d_H(J_c) < 1 + {|c|^2 \over4\log 2} + o(|c|^2).</math>

Since these are quasicircles corresponding to a dilatation

:<math> K=\sqrt{1+t\over 1-t}</math>

where

:<math> t= |1-\sqrt{1-4c}|,</math>

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

:<math>1+ 0.36 k^2\le d_H(C) \le 1 + 37 k^2.</math>

Having improved the lower bound following calculations for the Koch snowflake with Steffen Rohde and Oded Schramm, {{harvtxt|Astala|1994}} conjectured that

:<math> d_H(C) \le 1 + k^2.</math>

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

:<math>\lambda=d(2-d)\,\in (0,1)</math>

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

==Notes== {{reflist|2}}

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Category:Complex analysis Category:Dynamical systems Category:Fractals