In complex dynamics, 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 J ( f ) , {\displaystyle \operatorname {J} (f),} and the Fatou set is denoted F ( f ) . {\displaystyle \operatorname {F} (f).} 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.
Formal definition Let f ( z ) {\displaystyle f(z)} be a non-constant meromorphic function from the Riemann sphere onto itself. Such functions f ( z ) {\displaystyle f(z)} are precisely the non-constant complex rational functions, that is, f ( z ) = p ( z ) / q ( z ) {\displaystyle f(z)=p(z)/q(z)} where p ( z ) {\displaystyle p(z)} and q ( z ) {\displaystyle q(z)} 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 F 1 , . . . , F r {\displaystyle F_{1},...,F_{r}} that are left invariant by f ( z ) {\displaystyle f(z)} and are such that:
The union of the sets F i {\displaystyle F_{i}} is dense in the plane and
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