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Concept
Julia set
Summary
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") 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 f is commonly denoted and the Fatou set is denoted These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century.
Let be a non-constant holomorphic function from the Riemann sphere onto itself. Such functions are precisely the non-constant complex rational functions, that is, where and are complex polynomials. Assume that p and q have no common roots, and at least one has degree larger than 1. Then there is a finite number of open sets that are left invariant by and are such that:
The union of the sets is dense in the plane and
behaves in a regular and equal way on each of the sets .
The last statement means that the termini of the sequences of iterations generated by the points of are either precisely the same set, which is then a finite cycle, or they are finite cycles of circular or annular shaped sets that are lying concentrically. In the first case the cycle is attracting, in the second case it is neutral.
These sets are the Fatou domains of , and their union is the Fatou set of . Each of the Fatou domains contains at least one critical point of , that is, a (finite) point z satisfying , or if the degree of the numerator is at least two larger than the degree of the denominator , or if for some c and a rational function satisfying this condition.
The complement of is the Julia set of .
Official source
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