Dynamics in the Complex Bidisc

Let ∆n be the unit polydisc in Cn and let f be a holomorphic self map of ∆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 Cn. 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.