Iteration of holomorphic maps of the unit ball into itself

Let $\Omega$ be a plane disc and let $f$ be a holomorphic map of $\Omega$ into itself. It is known that the iterates ${f_n}$ of $f$ converge to a constant $\zeta \in \bar \Omega$ as $n \to \infty$ unless $f$ is a conformal map of $\Omega$ onto itself. In the present paper it is shown that a more complicated statement of this kind is true in the unit ball of ${{\mathbf{C}}^N}$.