### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 405-430 |

Number of pages | 26 |

Journal | Illinois Journal of Mathematics |

Volume | 49 |

Issue number | 2 |

State | Published - Jun 1 2005 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Illinois Journal of Mathematics*,

*49*(2), 405-430.

**On convergence to the Denjoy-wolff point.** / Bourdon, P. S.; Matache, V.; Shapiro, J. H.

Research output: Contribution to journal › Article

*Illinois Journal of Mathematics*, vol. 49, no. 2, pp. 405-430.

}

TY - JOUR

T1 - On convergence to the Denjoy-wolff point

AU - Bourdon, P. S.

AU - Matache, V.

AU - Shapiro, J. H.

PY - 2005/6/1

Y1 - 2005/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=15844420056&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=15844420056&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:15844420056

VL - 49

SP - 405

EP - 430

JO - Illinois Journal of Mathematics

JF - Illinois Journal of Mathematics

SN - 0019-2082

IS - 2

ER -