The theory of one-dimensional complex dynamical systems, understood as the global study of iteration of holomorphic mappings, has its roots in the early twentieth century with the work of Pierre Fatou and Gaston Julia. Local studies had been successfully attempted earlier, but it was Fatou and Julia's seminal work, inspired by Paul Montel's notion of normal families of mappings (then relatively new), that set the basis of what is known today as holomorphic dynamics. Both Fatou and Julia studied extensively the basic partition of the dynamical space into the two disjoint, completely invariant subsets, the Fatou set, which is the open set where tame dynamics occur – the set where Montel's normality appears – and its complement, the Julia set, which is the set of initial values whose orbits are chaotic.

Their greatest achievement, arguably, is their detailed description of the geometry and the dynamics of the connected components of the Fatou set, called the ‘Fatou components’. Their Classification Theorem asserts that every periodic Fatou component of a holomorphic map of the Riemann sphere (a rational map) is either (i) a component of an immediate basin of attraction of some attracting or parabolic cycle or (ii) a rotation domain conformally equivalent to a disc or an annulus.

Their work also left many interesting open questions, such as Fatou's No Wandering Domains Conjecture, which states that all Fatou components are eventually periodic, and waited some 60 years for its resolution.