On convergence to the Denjoy-Wolff point

For holomorphic selfmaps of the open unit disc U that are not elliptic automorphisms, the Schwarz Lemma and the Denjoy-Wolff Theorem combine to yield a remarkable result: each such map W has a (necessarily unique) "Denjoy-Wolff point" omega in the closed unit disc that attracts every orbit in the sense that the iterate sequence (phi([n])) converges to w uniformly on compact subsets of U. In this paper we prove that, except for the obvious counterexamples-inner functions having omega is an element of U-the iterate sequence exhibits an even stronger affinity for the Denjoy-Wolff point; phi([n]) -> to in the norm of the Hardy space HP for 1 <= p < infinity. For each such map, some subsequence of iterates converges to w almost everywhere on partial derivative U, and this leads us to investigate the question of almost-everywhere convergence of the entire iterate sequence. Here our work makes natural connections with two important aspects of the study of holomorphic selfmaps of the unit disc: linear- fractional models and ergodic properties of inner functions.