A central object of interest when describing the dynamics of a holomorphic self-map F of a complex manifold is the Fatou set of points with stable dynamical behaviour and its connected components, called Fatou components. A Fatou component is invariant, if F maps it inside itself. In one variable, an invariant Fatou component either admits a conjugation of F to a rotation or all its orbits accumulate at a single fixed point of F. In other words, the limit set has either full dimension 1 or trivial dimension 0.

One of the many new phenomena in dimension 2 is that the orbits of an invariant Fatou component may accumulate on a limit set of intermediate dimension 1. In this talk, I will present classification results on Fatou components with limit sets of dimension 1 and construct examples not arising as straightforward products of one-dimensional components in the categories of endomorphisms of projective space and automorphisms of \mathbb{C}^2 .