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 ${\displaystyle \operatorname {J} (f),}$ and the Fatou set is denoted ${\displaystyle \operatorname {F} (f).}$ 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.

Let ${\displaystyle f(z)}$ be a non-constant meromorphic function from the Riemann sphere onto itself. Such functions ${\displaystyle f(z)}$ are precisely the non-constant complex rational functions, that is, ${\displaystyle f(z)=p(z)/q(z)}$ where ${\displaystyle p(z)}$ and ${\displaystyle q(z)}$ 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 ${\displaystyle F_{1},...,F_{r}}$ that are left invariant by ${\displaystyle f(z)}$ and are such that:

The last statement means that the termini of the sequences of iterations generated by the points of ${\displaystyle F_{i}}$ 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 ${\displaystyle F_{i}}$ are the Fatou domains of ${\displaystyle f(z)}$, and their union is the Fatou set ${\displaystyle \operatorname {F} (f)}$ of ${\displaystyle f(z)}$. Each of the Fatou domains contains at least one critical point of ${\displaystyle f(z)}$, that is, a (finite) point z satisfying ${\displaystyle f'(z)=0}$, or ${\displaystyle f(z)=\infty }$ if the degree of the numerator ${\displaystyle p(z)}$ is at least two larger than the degree of the denominator ${\displaystyle q(z)}$, or if ${\displaystyle f(z)=1/g(z)+c}$ for some c and a rational function ${\displaystyle g(z)}$ satisfying this condition.

The complement of ${\displaystyle \operatorname {F} (f)}$ is the Julia set ${\displaystyle \operatorname {J} (f)}$ of ${\displaystyle f(z)}$. If all the critical points are preperiodic, that is they are not periodic but eventually land on a periodic cycle, then ${\displaystyle \operatorname {J} (f)}$ is all the sphere. Otherwise, ${\displaystyle \operatorname {J} (f)}$ is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers). Like ${\displaystyle \operatorname {F} (f)}$, ${\displaystyle \operatorname {J} (f)}$ is left invariant by ${\displaystyle f(z)}$, and on this set the iteration is repelling, meaning that ${\displaystyle |f(z)-f(w)|>|z-w|}$ for all w in a neighbourhood of z (within ${\displaystyle \operatorname {J} (f)}$). This means that ${\displaystyle f(z)}$ behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and they make up an infinitesimal part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.

There has been extensive research on the Fatou set and Julia set of iterated rational functions, known as rational maps. For example, it is known that the Fatou set of a rational map has either 0, 1, 2 or infinitely many components. Each component of the Fatou set of a rational map can be classified into one of four different classes.

The Julia set and the Fatou set of f are both completely invariant under iterations of the holomorphic function f: