On Convergence to the Denjoy-Wolff Point

Paul Bourdon, Valentin Matache, and Joel Shapiro

Illiinois J. Math. 49 (2005), 405--430.

Abstract: For holomorphic selfmaps of the open unit disc that are not elliptic automorphisms, the Schwarz Lemma and the Denjoy-Wolff Theorem combine to yield a remarkable result: each such map has a (necessarily unique) ``Denjoy-Wolff point'' in the closed unit disc that attracts every orbit. In this paper we prove that, except for the obvious counterexamples---inner functions having $\dwp\in\U$---the iterate sequence of the map actually converges to the Denjoy-Wolff point in the norm of the Hardy space $H^p$ for each finite p. This leads us to investigate the question of almost-everywhere convergence of iterate sequences. 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.

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Added 09/13/04: After we submitted this paper for publication we received a message from Pietro Poggi-Corradini informing us that he had employed more function-theoretic arguments to (independently) obtain the same a.e. convergence results we did for the interior fixed-point case, and the boundary fixed-point hyperbolic and parabolic automorphism cases. His paper can be downloaded at:

http://front.math.ucdavis.edu/math.CV/0407133