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Julia set
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{{Short description|Fractal sets in complex dynamics of mathematics}} {{More citations needed||date=July 2021}}[[File:Julia set, plotted with Matplotlib.svg|thumb|256px|A Julia set]] [[Image:Julia-Menge.Zoomfahrt.25p.webm|thumb|right|Zoom into a Julia set in the [[Complex number|complex-valued z-plane]] with the complex-valued [[Polynomial|polynomial function]] of [[Degree of a polynomial|second degree]]<br/><math>p(z) = z^2 + c</math><br/>and the parameters<br/>c<sub>re</sub> = c<sub>im</sub> = -0.5251993]] [[Image:Julia set 3d slice animation.ogg|thumb|256px|right|Three-dimensional slices through the (four-dimensional) Julia set of a function on the [[quaternion]]s]] <!-- "Fatou set" is also a Lemma of this article because "Fatou set" redirects here, thus it is bold below. Confusingly, the link behind "Fatou set" leads to somewhere else though. --> In [[complex dynamics]], the '''Julia set''' and the [[Classification of Fatou components|'''Fatou set''']] are two [[complement set|complementary sets]] (Julia "laces" and Fatou "dusts") defined from a [[function (mathematics)|function]]. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under [[iterated function|repeated iteration]] of the function, and the Julia set consists of values such that an arbitrarily small [[Perturbation theory|perturbation]] can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "[[chaos theory|chaotic]]". The Julia set of a function  {{mvar|f}}  is commonly denoted <math>\operatorname{J}(f),</math> and the Fatou set is denoted <math>\operatorname{F}(f).</math>{{efn|Regarding notation: For other branches of mathematics the notation <math>\operatorname{J}(f),</math> can also represent the [[Jacobian matrix#Jacobian matrix|Jacobian matrix]] of a real-valued mapping {{mvar|f}} between [[smooth manifold]]s.}} These sets are named after 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> whose work began the study of [[complex dynamics]] during the early 20th century.
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