Complex dynamics, or holomorphic dynamics, is the study of dynamical systems obtained by iterating a complex analytic mapping. This article focuses on the case of algebraic dynamics, where a polynomial or rational function is iterated. In geometric terms, that amounts to iterating a mapping from some algebraic variety to itself. The related theory of arithmetic dynamics studies iteration over the rational numbers or the p-adic numbers instead of the complex numbers.

A simple example that shows some of the main issues in complex dynamics is the mapping ${\displaystyle f(z)=z^{2}}$ from the complex numbers C to itself. It is helpful to view this as a map from the complex projective line ${\displaystyle \mathbf {CP} ^{1}}$ to itself, by adding a point ${\displaystyle \infty }$ to the complex numbers. (${\displaystyle \mathbf {CP} ^{1}}$ has the advantage of being compact.) The basic question is: given a point ${\displaystyle z}$ in ${\displaystyle \mathbf {CP} ^{1}}$, how does its orbit (or forward orbit)

behave, qualitatively? The answer is: if the absolute value |z| is less than 1, then the orbit converges to 0, in fact more than exponentially fast. If |z| is greater than 1, then the orbit converges to the point ${\displaystyle \infty }$ in ${\displaystyle \mathbf {CP} ^{1}}$, again more than exponentially fast. (Here 0 and ${\displaystyle \infty }$ are superattracting fixed points of f, meaning that the derivative of f is zero at those points. An attracting fixed point means one where the derivative of f has absolute value less than 1.)

On the other hand, suppose that ${\displaystyle |z|=1}$, meaning that z is on the unit circle in C. At these points, the dynamics of f is chaotic, in various ways. For example, for almost all points z on the circle in terms of measure theory, the forward orbit of z is dense in the circle, and in fact uniformly distributed on the circle. There are also infinitely many periodic points on the circle, meaning points with ${\displaystyle f^{r}(z)=z}$ for some positive integer r. (Here ${\displaystyle f^{r}(z)}$ means the result of applying f to z r times, ${\displaystyle f(f(\cdots (f(z))\cdots ))}$.) Even at periodic points z on the circle, the dynamics of f can be considered chaotic, since points near z diverge exponentially fast from z upon iterating f. (The periodic points of f on the unit circle are repelling: if ${\displaystyle f^{r}(z)=z}$, the derivative of ${\displaystyle f^{r}}$ at z has absolute value greater than 1.)

Pierre Fatou and Gaston Julia showed in the late 1910s that much of this story extends to any complex algebraic map from ${\displaystyle \mathbf {CP} ^{1}}$ to itself of degree greater than 1. (Such a mapping may be given by a polynomial ${\displaystyle f(z)}$ with complex coefficients, or more generally by a rational function.) Namely, there is always a compact subset of ${\displaystyle \mathbf {CP} ^{1}}$, the Julia set, on which the dynamics of f is chaotic. For the mapping ${\displaystyle f(z)=z^{2}}$, the Julia set is the unit circle. For other polynomial mappings, the Julia set is often highly irregular, for example a fractal in the sense that its Hausdorff dimension is not an integer. This occurs even for mappings as simple as ${\displaystyle f(z)=z^{2}+c}$ for a constant ${\displaystyle c\in \mathbf {C} }$. The Mandelbrot set is the set of complex numbers c such that the Julia set of ${\displaystyle f(z)=z^{2}+c}$ is connected.

There is a rather complete classification of the possible dynamics of a rational function ${\displaystyle f\colon \mathbf {CP} ^{1}\to \mathbf {CP} ^{1}}$ in the Fatou set, the complement of the Julia set, where the dynamics is "tame". Namely, Dennis Sullivan showed that each connected component U of the Fatou set is pre-periodic, meaning that there are natural numbers ${\displaystyle a<b}$ such that ${\displaystyle f^{a}(U)=f^{b}(U)}$. Therefore, to analyze the dynamics on a component U, one can assume after replacing f by an iterate that ${\displaystyle f(U)=U}$. Then either (1) U contains an attracting fixed point for f; (2) U is parabolic in the sense that all points in U approach a fixed point in the boundary of U; (3) U is a Siegel disk, meaning that the action of f on U is conjugate to an irrational rotation of the open unit disk; or (4) U is a Herman ring, meaning that the action of f on U is conjugate to an irrational rotation of an open annulus. (Note that the "backward orbit" of a point z in U, the set of points in ${\displaystyle \mathbf {CP} ^{1}}$ that map to z under some iterate of f, need not be contained in U.)

Complex dynamics has been effectively developed in any dimension. This section focuses on the mappings from complex projective space ${\displaystyle \mathbf {CP} ^{n}}$ to itself, the richest source of examples. The main results for ${\displaystyle \mathbf {CP} ^{n}}$ have been extended to a class of rational maps from any projective variety to itself. Note, however, that many varieties have no interesting self-maps.

Let f be an endomorphism of ${\displaystyle \mathbf {CP} ^{n}}$, meaning a morphism of algebraic varieties from ${\displaystyle \mathbf {CP} ^{n}}$ to itself, for a positive integer n. Such a mapping is given in homogeneous coordinates by