Julia set

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In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small 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 'chaotic'.

The Julia set of a function ƒ is commonly denoted J(ƒ), and the Fatou set is denoted F(ƒ).[1] These sets are named after the French mathematicians Gaston Julia[2] and Pierre Fatou,[3] whose work began the study of complex dynamics during the early 20th century.

Contents

Formal definition

Let f(z) be a complex rational map from the plane into itself, that is, f(z) = p(z) / q(z), where p(z) and q(z) are complex polynomials. Then there are a finite number of open sets F_i, i = 1, \dots, r, that are left invariant by f(z) and are such that:

The last statement means that the termini of the sequences of iterations generated by the points of Fi are either precisely the same set, which is then a finite cycle, or they are finite cycles of finite or annular shaped sets that are lying concentrically. In the first case the cycle is attracting, in the second it is neutral.

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