Peano-Russell notation

2011-06-09T00:06:10Z

24.44.161.44: Changed the dots so that they are different, per Russel's original notation

Siegel disc is a connected [[Classification of Fatou components|component in the Fatou set]] where the dynamics is analytically [[Topological_conjugation|conjugated]] to an [[irrational rotation]].

==Description==
Given a [[holomorphic]] [[endomorphism]] <math>f:S\to S</math> on a [[Riemann surface]] <math>S</math> we consider the [[dynamical system]] generated by the [[Iterated function|iterates]] of <math>f</math> denoted by <math>f^n=f\circ\stackrel{\left(n\right)}{\cdots}\circ f</math>. We then call the [[Orbit_(dynamics)|orbit]] <math>\mathcal{O}^+(z_0)</math> of <math>z_0</math> as the set of forward iterates of <math>z_0</math>. We are interested in the asymptotic behavior of the orbits in <math>S</math> (which will usually be <math>\mathbb{C}</math>, the [[complex plane]] or <math>\mathbb{\hat C}=\mathbb{C}\cup\{\infty\}</math>, the [[Riemann sphere]]), and we call <math>S</math> the [[phase plane]] or ''dynamical plane''.

One possible asymptotic behavior for a point <math>z_0</math> is to be a [[fixed point (mathematics)|fixed point]], or in general a ''periodic point''. In this last case <math>f^p(z_0)=z_0</math> where <math>p</math> is the [[Orbit_(dynamics)|period]] and <math>p=1</math> means <math>z_0</math> is a fixed point. We can then define the ''multiplier'' of the orbit as <math>\rho=(f^p)'(z_0)</math> and this enables us to classify periodic orbits as ''attracting'' if <math>|\rho|<1</math> ''superattracting'' if <math>|\rho|=0</math>), ''repelling'' if <math>|\rho|>1</math> and indifferent if <math>\rho=1</math>. Indifferent periodic orbits split in ''rationally indifferent'' and ''irrationally indifferent'', depending on whether <math>\rho^n=1</math> for some <math>n\in\mathbb{Z}</math> or <math>\rho^n\neq1</math> for all <math>n\in\mathbb{Z}</math>, respectively.

'''Siegel discs''' are one of the possible cases of connected components in the Fatou set (the complementary set of the [[Julia set]]), according to [[Classification of Fatou components]], and can occur around irrationally indifferent periodic points. The Fatou set is, roughly, the set of points where the iterates behave similarly to their neighbours (they form a [[normal family]]). '''Siegel discs''' correspond to points where the dynamics of <math>f</math> is analytically
[[Topological_conjugation|conjugated]] to an irrational rotation of the complex disc.

==Name==
The disk is named in honor of [[Carl Ludwig Siegel]].
==Gallery==
<gallery widths="300px" heights="300px" perrow=3>
Image:SiegelDisk.jpg |Siegel disc for a polynomial-like mapping
Image:FigureJuliaSetForPolynomialLike.jpg|Julia set for <math>B(z)=\lambda a(e^{z/a}(z+1-a)+a-1)</math>, where <math>a=15-15i</math> and <math>\lambda</math> is the [[golden ratio]]. Orbits of some points inside the '''Siegel disc''' emphasized
Image:UnboundedSiegeldisk.jpg|Julia set for <math>B(z)=\lambda a(e^{z/a}(z+1-a)+a-1)</math>, where <math>a=-0.33258+0.10324i</math> and <math>\lambda</math> is the [[golden ratio]]. Orbits of some points inside the '''Siegel disc''' emphasized. The Siegel disc is either unbounded or its boundary is an indecomposable continuum.<ref>Rubén Berenguel and Núria Fagella ''An entire transcendental family with a persistent Siegel disc, 2009 preprint: [http://arxiv.org/abs/0907.0116 arXiV:0907.0116]</ref>

File:Golden Mean Quadratic Siegel Disc Speed.png | Filled Julia set for <math>f_c(z) = z*z + c</math> for [[Golden ratio|Golden Mean]] rotation number with interior colored propotional to the average discrete velocity on the orbit = abs( z_(n+1) - z_n ). Note that there is only one Siegel disc and many preimages of the orbits within the Siegel disk
File:Golden Mean Quadratic Siegel Disc.png|Filled Julia set for <math>f_c(z) = z*z + c</math> for [[Golden ratio|Golden Mean]] rotation number with Siegel disc and some orbits inside
File:Siegel quadratic 3,2,1000,1... ,.png|Julia set of quadratic polynomial with Siegel disk for rotation number [3,2,1000,1...]
</gallery>

==Formal definition==
Let <math>f:S\to S</math> be a [[holomorphic]] [[endomorphism]] where <math>S</math> is a [[Riemann surface]], and let U be a [[connected component (analysis)| connected component]] of the Fatou set <math>\mathcal{F}(f)</math>. We say U is a Siegel disc of f around the point z_0 if there exists an analytic homeomorphism <math>\phi:U\to\mathbb{D}</math> where <math>\mathbb{D}</math> is the unit disc and such that <math>\phi(f^n(\phi^{-1}(z)))=e^{2\pi i\alpha}z</math> for some <math>\alpha\in\mathbb{R}\backslash\mathbb{Q}</math> and <math>\phi(z_0)=0</math>.

[[Carl Ludwig Siegel|Siegel's]] theorem proves the existence of '''Siegel discs''' for [[Irrational Number|irrational numbers]] satisfying a ''strong irrationality condition'' (a [[Diophantine condition]]), thus solving an open problem since Fatou conjectured his theorem on the [[Classification of Fatou components]].<ref>[[Lennart Carleson]] and Theodore W. Gamelin, ''Complex Dynamics'', Springer 1993</ref>

Later [[A. D. Brjuno]] improved this condition on the irrationality, enlarging it to the [[Brjuno number]]s.<ref name="MilnorComplexDynamics">[[John W. Milnor]], ''Dynamics in One Complex Variable'' (Third Edition), Annals of Mathematics Studies 160, Princeton University Press 2006 (First appeared in 1990 as a [http://www.math.sunysb.edu/preprints.html Stony Brook IMS Preprint], available as [http://www.arxiv.org/abs/math.DS/9201272 arXiV:math.DS/9201272].)</ref>

This is part of the result from the [[Classification of Fatou components]].

==See also==
* [[Herman ring]]
{{wikibooks|Fractals/Iterations in the complex plane/siegel}}

==References==
{{reflist}}
* [http://www.scholarpedia.org/article/Siegel_disks Siegel disks ar Scholarpedia]

[[Category:Fractals]]
[[Category:Limit sets]]
[[Category:Complex dynamics]]

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Peano-Russell notation