A generalization of a theorem of Heins

Abstract:

Let $\mathcal {H}(\Delta , \Delta )$ be the family of holomorphic selfmaps of the unit disk $\Delta$ in the complex plane $C$. Heins established the continuity of the functional $\psi$ which assigns to $f \in \overline {{\mathcal {H}}(\Delta , \Delta )}-\{id\}$ ($id$ denotes the identity map) either (i) the fixed point of $f$ or (ii) the limit of its iterations or (iii) $f(\Delta )$ if $f(\Delta ) \cap \partial \Delta \not = \emptyset$ ($\partial \Delta$ represents the boundary of $\Delta$). Using an Abate extension of the Denjoy-Wolff lemma to strongly convex domains, we extend this result of Heins to selfmaps of strongly convex domains in $C^{n}$ with $C^{2}$ boundary.