Dynamics in the complex bidisc

Abstract

Let Delta^{n} be the unit polydisc in C^{n} and let f be a holomorphic self map of Delta^{n}. When n=1, it is well known, by Schwarz's lemma, that f has at most one fixed point in the unit disc. If no such point exists then f has a unique boundary point, call it x, such that every horocycle E(x,R) of center x and radius R>0 is sent into itself by f. This boundary point is called the "Wolff point of f". In this paper we propose a definition of Wolff points for holomorphic maps defined on a bounded domain of C^{n}. In particular we characterize the set of Wolff points, W(f), of a holomorphic self-map f of the bidisc in terms of the properties of the components of the map f itself.