Attractors in ℙ²

Summary

We investigate attractors for holomorphic maps from ℙ ^k→ ℙ ^k, emphasizing the case k = 2. The interest in attractors stems from the fact that when a map is subject to small random perturbations, the long-term dynamics of the resulting system live near the map's attractors. In the case k = 1, that is, the case of rational functions on the Riemann sphere, the attractors are either periodic orbits or the whole sphere. In higher dimensions, however, there are other possibilities, which we call nontrivial. In addition to giving some examples of nontrivial attractors, we prove some general results about such attractors inℙ ², among them that a given map can have at most one nontrivial attractor K, that K is then connected, has pseudoconvex complement, and contains a nonconstant entire image of ℂ, and that an attractor for a map f is also an attractor for any iterate fⁿ.

1. Introduction

We recall first some general notions from the theory of dynamical systems. See [Ruelle 1989] for background.