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Julia set
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==Equivalent descriptions of the Julia set== * <math>\operatorname{J}(f)</math> is the smallest closed set containing at least three points which is completely invariant under ''f''. * <math>\operatorname{J}(f)</math> is the [[Closure (topology)|closure]] of the set of repelling [[periodic point]]s. * 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.) * If ''f'' is an [[entire function]], then <math>\operatorname{J}(f)</math> is the [[boundary (topology)|boundary]] of the set of points which converge to infinity under iteration. * If ''f'' is a polynomial, then <math>\operatorname{J}(f)</math> is the boundary of the [[filled Julia set]]; that is, those points whose orbits under iterations of ''f'' remain bounded.
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