Julia set
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2008-07-05T15:57:20Z
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The image is arguable, but the video most certainly doesn't need to be a thumb
[[Image:Julia set (ice).png|256px|right]]
[[Image:Julia set 3d slice animation.ogg|256px|right]]
In [[complex dynamics]], the '''Julia set''' <math>J(f)\,</math> of a [[holomorphic function]] <math>f\,</math> informally consists of those points whose long-time behavior under [[iterated function|repeated iteration]] of <math>f\,</math> can change drastically under arbitrarily small perturbations.
The '''Fatou set''' <math>F(f)\,</math> of <math>f\,</math> is the [[complement (mathematics)|complement]] of the Julia set: that is, the set of points which exhibit 'stable' behavior.
Thus on <math>F(f)\,</math>, the behavior of <math>f\,</math> is 'regular', while on <math>J(f)\,</math>, it is '[[chaos theory|chaotic]]'.
These sets are named in honor of the French mathematicians [[Gaston Julia]]<ref>Gaston Julia (1918) "Mémoire sur l'iteration des fonctions rationnelles," ''Journal de Mathématiques Pures et Appliquées'', vol. 8, pages 47-245.</ref> and [[Pierre Fatou]]<ref>Pierre Fatou (1917) "Sur les substitutions rationnelles," ''Comptes Rendus de l'Académie des Sciences de Paris'', vol. 164, pages 806-808 and vol. 165, pages 992-995.</ref>, who initiated the theory of [[complex dynamics]] in the early 20th century.
==Formal definition==
Let
:<math>f:X\to X\,</math>
be an analytic self-map of a [[Riemann surface]] <math>X\,</math>. We will assume that <math>X\,</math> is either the [[Riemann sphere]], the [[complex plane]], or the once-punctured complex plane, as the other cases do not give rise to interesting dynamics. (Such maps are [[Picard theorem|completely classified]].)
We will be considering <math>f\,</math> as a [[dynamical system|discrete dynamical system]] on the [[phase space]]
<math>X\,</math>, so we are interested in the behavior of the [[iterated function|iterates]] <math>f^n\,</math> of <math>f\,</math> (that is, the <math>n\,</math>-fold compositions of <math>f\,</math> with itself).
The Fatou set of <math>f\,</math> consists of all points <math>z\in X\,</math> such that the family of iterates
:<math>(f^n)_{n\in\mathbb{N}}</math>
forms a [[normal family]] in the sense of [[Paul Antoine Aristide Montel|Montel]] when restricted to some open neighborhood of <math>z\,</math>.
The Julia set of <math>f\,</math> is the complement of the Fatou set in <math>X\,</math>.
==Equivalent descriptions of the Julia set==
<ul>
<li> <math>J(f)\,</math> is the smallest closed set containing at least three points which is completely invariant under <math>f\,</math>.
<li> <math>J(f)\,</math> is the [[Closure (mathematics)|closure]] of the set of repelling [[periodic point]]s.
<li> For all but at most two points <math>z\in X\,</math>, the Julia set is the set of limit points of the full backwards orbit <math>\bigcup_n f^{-n}(z)</math>. (This suggests a simple algorithm for plotting Julia sets, see below.)
<li> If <math>f\,</math> is an [[entire function]] - in particular, when <math>f\,</math> is a [[polynomial]], then <math>J(f)\,</math> is the [[boundary (topology)|boundary]] of the set of points which converge to infinity under iteration.
<li> If <math>f\,</math> is a polynomial, then <math>J(f)\,</math> is the boundary of the [[filled Julia set]]; that is, those points whose orbits under <math>f\,</math> remain bounded.
</ul>
==Properties of the Julia set and Fatou set==
The Julia set and the Fatou set of <math>f</math> are both [[Invariant (mathematics)|completely invariant]] under <math>f</math>, i.e.
<math>\ f^{-1}(J(f)) = f(J(f)) = J(f)</math>
and
<math>\ f^{-1}(F(f)) = f(F(f)) = F(f)</math>.
<ref>Beardon, ''Iteration of Rational Functions'', Theorem 3.2.4</ref>
== Rational maps ==
[[Image:Julia-set N z3-1.png||thumb|right|Julia set (in white) for the rational function associated to [[Newton method|Newton's method]] for ƒ:''z''→''z''<sup>3</sup>−1. Coloring of Fatou set according to attractor (the roots of ƒ)]]
There has been extensive research on the Fatou set and Julia set of iterated [[rational functions]], known as rational maps. For example, it is known that the Fatou set of a rational map has either 0,1,2 or infinitely many [[Connected component (analysis)|components]].<ref>Beardon, ''Iteration of Rational Functions'', Theorem 5.6.2</ref> Each component of the Fatou set of a rational map can be classified into one of [[Classification of Fatou components|four different classes]].<ref>Beardon, Theorem 7.1.1</ref>
==Quadratic polynomials==
{{Wikibooks|Fractals }}
A very popular complex dynamical system is given by the family of [[Complex quadratic polynomial| quadratic polynomials]], a special case of [[Rational function| rational map]]s. The [[Complex quadratic polynomial| quadratic polynomials]] can be expressed as
:<math>f_c(z) = z^2 + c\,</math>
(where <math>c\,</math> is a complex parameter).
<gallery>
Image:Time escape Julia set from coordinate (phi-2, 0).jpg|Filled Julia set for f<sub>c</sub>, c=1−φ where φ is the [[golden ratio]]
Image:Time escape Julia set from coordinate (phi-2, phi-1).jpg|Julia set for f<sub>c</sub>, c=(φ−2)+(φ−1)i =-0.4+0.6i
Image:Time escape Julia set from coordinate (0.285, 0).jpg|Julia set for f<sub>c</sub>, c=0.285+0i
Image:Julia set (highres 01).jpg|Julia set for f<sub>c</sub>, c=0.285+0.01i
Image:Julia_set_camp3.jpg|Julia set for f<sub>c</sub>, c=0.45+0.1428i
Image:Julia_set_camp1.jpg|Julia set for f<sub>c</sub>, c=-0.70176-0.3842i
Image:Julia_set_camp2.jpg|Julia set for f<sub>c</sub>, c=-0.835-0.2321i
Image:Julia_set_camp4_hi_rez.png|Julia set for f<sub>c</sub>, c=-0.8+0.156i
</gallery>
[[Image:725 Julia sets.png|thumb|A Julia set plot showing julia sets for different values of c, the plot resembles the [[Mandelbrot set]]]]
The parameter plane of quadratic polynomials - that is, the plane of possible <math>c</math>-values - gives rise to the famous [[Mandelbrot set]]. Indeed, the Mandelbrot set is defined as the set of all <math>c</math> such that <math>J(f_c)\,</math> is [[connected set|connected]]. For parameters outside the Mandelbrot set, the Julia set is a [[Cantor set]]: in this case it is sometimes referred to as '''Fatou dust'''.
In many cases, the Julia set of ''c'' looks like the Mandelbrot set in sufficiently small neighborhoods of ''c''. This is true, in particular, for so-called [[Misiurewicz point|'Misiurewicz' parameters]], i.e. parameters <math>c</math> for which the critical point is pre-periodic. For instance:
*At ''c''= i, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
*At ''c'' = −2, the tip of the long spiky tail, the Julia set is a straight line segment.
In other words the Julia sets <math>J(f_c)\,</math> are locally similar around [[Misiurewicz point]]s.<ref>[http://projecteuclid.org/euclid.cmp/1104201823 Lei.pdf] Tan Lei, "Similarity between the Mandelbrot set and Julia Sets", Communications in Mathematical Physics 134 (1990), pp. 587-617.</ref>
==Generalizations==
The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their domain; most notably [[meromorphic function|transcendental meromorphic functions]] and Epstein's 'finite-type maps'.
Julia sets are also commonly defined in the study of dynamics in several complex variables.
== Plotting the Julia set ==
===using backwards (inverse) iteration ('''IIM''' ) ===
[[Image:Reversed Julia set C = ( 0.4 0.3 ).gif|thumb|A Julia set plot, generated using backwards iteration]]
As mentioned above, the Julia set can be found as the set of limit points of the set of pre-images of (essentially) any given point. So we can try to plot the Julia set of a given function as follows. Start with any point <math>z\,</math> we know to be in the Julia set, such as a repelling periodic point, and compute all pre-images of <math>z\,</math> under some high iterate <math>f^n\,</math> of <math>f\,</math>.
Unfortunately, as the number of iterated pre-images grows exponentially, this is not computationally feasible. However, we can adjust this method, in a similar way as the "random game" method for
[[iterated function system]]s. That is, in each step, we choose at random one of the inverse images of <math>f\,</math>.
For example, for the quadratic polynomial <math>f_c\,</math>, the backwards iteration is described by
:<math>z_{n+1}^2 = z_n - c.</math>
At each step, one of the two square roots is selected at random.
Note that certain parts of the Julia set are quite hard to reach with the reverse Julia algorithm. For this reason, other methods usually produce better images.
=== using '''DEM/J''' ===
[[Image:Demj.jpg|thumb|right|Julia set : image with C source code using DEM/J]]
==See also==
* [[Limit set]]
* [[Stable and unstable sets]]
* [[No wandering domain theorem]]
* [[Classification of Fatou components|Fatou components]]
{{commons|Julia set}}
==References==
<references/>
* [[Lennart Carleson]] and Theodore W. Gamelin, ''Complex Dynamics'', Springer 1993
* Adrien Douady and John H. Hubbard, "Etude dynamique des polynômes complexes", ''Prépublications mathémathiques d'Orsay'' 2/4 (1984 / 1985)
* 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].)
* Alexander Bogomolny, "[http://www.cut-the-knot.org/Curriculum/Algebra/JuliaIndexing.shtml Mandelbrot Set and Indexing of Julia Sets]" at ''[[cut-the-knot]]''.
* Evgeny Demidov, "[http://ibiblio.org/e-notes/MSet/Contents.htm The Mandelbrot and Julia sets Anatomy]" (2003)
* Alan F. Beardon, ''Iteration of Rational Functions'', Springer 1991, ISBN 0-387-95151-2
==Links==
*http://mathmo.blogspot.com/2007/04/essay-backtrack-julia-sets.html
*http://mcgoodwin.net/julia/juliajewels.html
*http://www.lucypringle.co.uk/photos/1996/uk1996ck.shtml Crop circle Julia set
{{Spoken Wikipedia|Julia_Set.ogg|2007-06-18}}
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