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 $\phi$ has a (necessarily unique) ``Denjoy-Wolff
point'' $\dwp$ in the closed unit disc that attracts every
orbit in the sense that the iterate sequence $(\phin)$
converges to $\dwp$ uniformly on compact subsets of $\U$. In
this paper we prove that, except for the obvious
counterexamples—inner functions having $\dwp\in\U$—the
iterate sequence exhibits an even stronger affinity for the
Denjoy-Wolff point; $\phin\goesto\dwp$ in the norm of the
Hardy space $H^p$ for $1\le p<\infty$. For each such map,
some subsequence of iterates converges to $\dwp$ almost
everywhere on $\bdu$, 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.