# Quasicircle

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{{Short description|Quasiconformal complex image of a circle}}
{{distinguish|Pseudocircle}}
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In [mathematics](/source/mathematics), a '''quasicircle''' is a [Jordan curve](/source/Jordan_curve) in the [complex plane](/source/complex_plane) that is the image of a [circle](/source/circle) under a [quasiconformal mapping](/source/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 [arc](/source/arc_(geometry))s.<ref>{{harvnb|Lehto|Virtanen|1973}}</ref><ref>{{harvnb|Krzyz|1983|p=49}}</ref> In [complex analysis](/source/complex_analysis) and [geometric function theory](/source/geometric_function_theory), quasicircles play a fundamental role in the description of the  [universal Teichmüller space](/source/universal_Teichm%C3%BCller_space), through [quasisymmetric homeomorphism](/source/quasisymmetric_map)s of the circle. Quasicircles also play an important role in [complex dynamical system](/source/complex_dynamical_system)s.

==Definitions==
A quasicircle is defined as the image of a circle under a [quasiconformal mapping](/source/quasiconformal_mapping) of the [extended complex plane](/source/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](/source/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](/source/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 curve](/source/Jordan_curve)s for which the absolute value of the [cross-ratio](/source/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''<sub>1</sub> and ''z''<sub>2</sub> are chosen on the curve and ''z''<sub>3</sub> 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''<sub>1</sub>, ''z''<sub>2</sub> are any points on the curve and  ''z''<sub>3</sub> 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](/source/Lipschitz_continuity) ''f'', i.e. satisfying

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

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

==Quasicircles and quasisymmetric homeomorphisms==
If φ is a [quasisymmetric homeomorphism](/source/quasisymmetric_map) 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](/source/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 transformation](/source/M%C3%B6bius_transformation)s provides a model of [universal Teichmüller space](/source/universal_Teichm%C3%BCller_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](/source/Holomorphic_function) [univalent function](/source/univalent_function)s ''f''(''z'') on the unit disk ''D''. Let Ω = ''f''(''D'').  As Carathéodory had proved using his theory of [prime end](/source/prime_end)s, ''f'' extends continuously to the unit circle [if and only if](/source/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](/source/Carath%C3%A9odory's_theorem_(conformal_mapping)) 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](/source/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''<sub>1</sub> 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](/source/Koch_snowflake)]]
Quasicircles were known to arise as the [Julia set](/source/Julia_set)s of rational maps ''R''(''z''). {{harvtxt|Sullivan|1985}} proved that if the [Fatou set](/source/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''<sup>2</sup> + ''c'' with an attracting fixed point
* the [Douady rabbit](/source/Douady_rabbit) (''c'' = –0.122561 + 0.744862i, where ''c''<sup>3</sup> + 2 ''c''<sup>2</sup> + ''c'' + 1 = 0)
* quadratic polynomials ''z''<sup>2</sup> + λ''z'' with |λ| < 1
* the [Koch snowflake](/source/Koch_snowflake)

==Quasi-Fuchsian groups==
[Quasi-Fuchsian group](/source/Quasi-Fuchsian_group)s are obtained as quasiconformal deformations of [Fuchsian group](/source/Fuchsian_group)s. By definition their [limit set](/source/limit_set)s 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](/source/Riemann_surface) ''D'' /  Γ.)

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

The [Beltrami equation](/source/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](/source/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](/source/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 set](/source/Julia_set)s ''J''<sub>c</sub> of the iterates of the [rational map](/source/rational_map)s

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

had been estimated as result of the work of [Rufus Bowen](/source/Rufus_Bowen) and [David Ruelle](/source/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](/source/Koch_snowflake) with Steffen Rohde and [Oded Schramm](/source/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](/source/hyperbolic_3-manifold).<ref>{{harvnb|Astala|Zinsmeister|1994}}</ref><ref>{{harvnb|Marden|2007|p=284}}</ref>

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

==References==
*{{citation|last=Ahlfors|first=Lars V.|authorlink=Lars Ahlfors|title=Lectures on quasiconformal mappings|publisher=Van Nostrand|year=1966}}
*{{citation | title=Quasiconformal reflections | last= Ahlfors|first=L. | authorlink=Lars Ahlfors | journal=[Acta Mathematica](/source/Acta_Mathematica) | volume=109 | year=1963 | pages=291–301 | zbl=0121.06403 | doi=10.1007/bf02391816| doi-access=free }}
*{{citation|last=Astala|first= K.|title=Distortion of area and dimension under quasiconformal mappings in the plane|journal=Proc. Natl. Acad. Sci. U.S.A. |volume=90 |issue= 24|year=1993|pages=11958–11959|doi=10.1073/pnas.90.24.11958|pmid= 11607447|pmc=48104|bibcode=1993PNAS...9011958A|doi-access= free}}
*{{citation|last1=Astala|first1= K.|last2=Zinsmeister|first2= M.|title=Holomorphic families of quasi-Fuchsian groups|journal=Ergodic Theory Dynam. Systems|volume= 14|issue= 2|year=1994| pages=207–212|doi=10.1017/s0143385700007847|s2cid= 121209816}}
*{{citation|last=Astala|first=K.|title=Area distortion of quasiconformal mappings|journal=Acta Math.|volume= 173 |year=1994|pages= 37–60|doi=10.1007/bf02392568|doi-access=free}}
*{{citation|title=Elliptic partial differential equations and quasiconformal mappings in the plane|volume= 48|series= Princeton mathematical series|
first1=Kari|last1= Astala|first2= Tadeusz |last2=Iwaniec|author2-link=Tadeusz Iwaniec|first3= Gaven|last3= Martin|author3-link=Gaven Martin|publisher=Princeton University Press|year= 2009|
isbn=978-0-691-13777-3|pages=332–342}}, Section 13.2, Dimension of quasicircles.
*{{citation|last1=Becker|first1= J.|last2= Pommerenke|first2= C.|title=On the Hausdorff dimension of quasicircles|journal=Ann. Acad. Sci. Fenn. Ser. A I Math. |volume=12 |year=1987|pages= 329–333|doi= 10.5186/aasfm.1987.1206|doi-access=free}}
* {{cite journal |last=Bers |first=Lipman |author-link=Lipman Bers |date=August 1961 |title=Uniformization by Beltrami equations |journal=Communications on Pure and Applied Mathematics |volume=14 |number=3 |pages=215–228 |doi=10.1002/cpa.3160140304}}
*{{citation|last=Bowen|first= R.|title=Hausdorff dimension of quasicircles|journal=Inst. Hautes Études Sci. Publ. Math.|volume= 50|year=1979|pages= 11–25|doi= 10.1007/BF02684767|s2cid= 55631433|url= http://www.numdam.org/item/PMIHES_1979__50__11_0/}}
*{{citation|last1=Carleson|first1=L.|author-link=Lennart Carleson|last2=Gamelin|first2=T. D. W.|author-link2=Theodore Gamelin|title=Complex dynamics|series=Universitext: Tracts in Mathematics|publisher=Springer-Verlag|year=1993|isbn=978-0-387-97942-7|url-access=registration|url=https://archive.org/details/complexdynamics0000carl}}
*{{citation|last1=Gehring|first1= F. W.|last2= Väisälä|first2= J.|title=Hausdorff dimension and quasiconformal mappings|journal=Journal of the London Mathematical Society|volume= 6|issue= 3|year= 1973|pages= 504–512|doi=10.1112/jlms/s2-6.3.504|citeseerx= 10.1.1.125.2374}}
*{{citation|last=Gehring|first= F. W.|title=Characteristic properties of quasidisks|series=Séminaire de Mathématiques Supérieures |volume=84 |publisher=Presses de l'Université de Montréal|year= 1982|isbn= 978-2-7606-0601-2}}
*{{citation|first1=Y. |last1=Imayoshi|first2=M.|last2=Taniguchi|title=An Introduction to Teichmüller spaces|publisher=Springer-Verlag|year=1992|isbn=978-0-387-70088-5}}
*{{citation|first=O.|last=Lehto|title=Univalent functions and Teichmüller spaces|publisher=Springer-Verlag|year=1987|isbn=978-0-387-96310-5|pages=50–59, 111–118, 196–205}}
* {{cite book |last1=Krzyz |first1=J. |title=Quasiconformal Mappings in the Plane: Parametrical Methods |date=1983 |publisher=Springer Berlin / Heidelberg |location=Berlin, Heidelberg |isbn=978-3540119890}}
*{{citation|last1=Lehto|first1=O.|last2= Virtanen|first2= K. I.|title=Quasiconformal mappings in the plane|edition=Second|series= Die Grundlehren der mathematischen Wissenschaften|volume= 126|publisher=Springer-Verlag|year= 1973}}
*{{citation|last=Marden|first= A.|title=Outer circles. An introduction to hyperbolic 3-manifolds|publisher= Cambridge University Press|year= 2007|isbn= 978-0-521-83974-7}}
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*{{citation|last=Pfluger|first= A.|title=Ueber die Konstruktion Riemannscher Flächen durch Verheftung|journal=J. Indian Math. Soc.|volume= 24 |year=1961 |pages=401–412}}
*{{citation|last=Pommerenke|first= C.|authorlink=Christian Pommerenke|title=Univalent functions, with a chapter on quadratic differentials by Gerd Jensen|series= Studia Mathematica/Mathematische Lehrbücher|volume=15|publisher= Vandenhoeck & Ruprecht|year= 1975}}
*{{citation|last=Rohde|first= S.|title=On conformal welding and quasicircles|journal=Michigan Math. J. |volume=38 |year=1991|pages=111–116|doi=10.1307/mmj/1029004266|doi-access=free}}
*{{citation|last=Sullivan|first= D.|title=Discrete conformal groups and measurable dynamics|journal=Bull. Amer. Math. Soc.|volume= 6 |year=1982|pages=57–73|doi=10.1090/s0273-0979-1982-14966-7|doi-access=free}}
*{{citation|last=Sullivan|first=D.|title=Quasiconformal homeomorphisms and dynamics, I, Solution of the Fatou-Julia problem on wandering domains|journal= Annals of Mathematics|year= 1985|volume=122|issue=2|pages= 401–418|doi=10.2307/1971308|jstor=1971308}}
*{{citation|last=Tienari|first= M.|title=Fortsetzung einer quasikonformen Abbildung über einen Jordanbogen|journal=Ann. Acad. Sci. Fenn. Ser. A|volume= 321|year= 1962}}
*{{citation |first=S.|last= Smirnov | authorlink=Stanislav Smirnov | title=Dimension of quasicircles | journal=[Acta Mathematica](/source/Acta_Mathematica) | volume=205 | year=2010 | pages=189–197 | mr=2736155 | doi=10.1007/s11511-010-0053-8 | arxiv=0904.1237 |s2cid= 17945998 }}

Category:Complex analysis
Category:Dynamical systems
Category:Fractals

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