On convergence to the Denjoy-wolff point

Abstract

For holomorphic selfmaps of the open unit disc double-struck U sign that are not elliptic automorphisms, the Schwarz Lemma and the Denjoy-Wolff Theorem combine to yield a remarkable result: each such map φ has a (necessarily unique) "Denjoy-Wolff point" ω in the closed unit disc that attracts every orbit in the sense that the iterate sequence (φ [n]) converges to ω uniformly on compact subsets of double-struck U sign. In this paper we prove that, except for the obvious counterexamples - inner functions having ω ∈ double-struck U sign - the iterate sequence exhibits an even stronger affinity for the Denjoy-Wolff point; φ[n] → ω in the norm of the Hardy space H p for 1 ≤ p < ∞. For each such map, some subsequence of iterates converges to ω almost everywhere on ∂double-struck U sign, 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. © 2005 University of Illinois.