Peano-Russell notation: Difference between revisions

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== For that reason ｛key｝ ==


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]].
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==Description==
  <li>[http://touchscreengenius.carolasmith.com/forum/profile.php?id=3949 http://touchscreengenius.carolasmith.com/forum/profile.php?id=3949]</li>
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''.  
 
 
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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.
 
 
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'''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.
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==Name==
  <li>[http://general.assembly.codesria.org/spip.php?article87&lang=pt/ http://general.assembly.codesria.org/spip.php?article87&lang=pt/]</li>
The disk is named in honor of [[Carl Ludwig Siegel]].
 
==Gallery==
</ul>
<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]]

Latest revision as of 13:50, 23 November 2014

For that reason ｛key｝

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The second reason articles are so helpful is that search engines LOVE good content, which they will find in your articles. There's no better way to help your website increase in the search engine rankings than supplying great happy to your website.

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Russia was one of the few countries that refused to change to the Gregorian calendar in 1582 and continued to celebrate its holidays as before. The Old New Year is celebrated on January 14. This record's for children like us, who just love music, because it saves their lives every single day," McCracken says. "At the end of your day, I'm the only one who has to live and die with myself.

Seventeen people on twelve sledges pulled by 160 dogs took a stressful threemonth trip up to Bennet Island, where they found the diaries and also the expedition collections, which shed light on the tragic fate of Baron Eduard Von Toll and the companions. Kolchak's fiance soon traveled to satisfy him in Siberia and immediately after the wedding he left to Port Arthur, where the first battles took place.

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