On convergence to the Denjoy-wolff point

P. S. Bourdon, Valentin Matache, J. H. Shapiro

Research output: Contribution to journalArticle

15 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)405-430
Number of pages26
JournalIllinois Journal of Mathematics
Volume49
Issue number2
StatePublished - Jun 1 2005

Fingerprint

Iterate
Unit Disk
Inner Functions
Schwarz Lemma
Converge
Almost Everywhere Convergence
Hardy Space
Subsequence
Affine transformation
Counterexample
Automorphisms
Fractional
Orbit
Entire
Norm
Closed
Subset
Theorem
Model

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Bourdon, P. S., Matache, V., & Shapiro, J. H. (2005). On convergence to the Denjoy-wolff point. Illinois Journal of Mathematics, 49(2), 405-430.

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

In: Illinois Journal of Mathematics, Vol. 49, No. 2, 01.06.2005, p. 405-430.

Research output: Contribution to journalArticle

Bourdon, PS, Matache, V & Shapiro, JH 2005, 'On convergence to the Denjoy-wolff point', Illinois Journal of Mathematics, vol. 49, no. 2, pp. 405-430.
Bourdon, P. S. ; Matache, Valentin ; Shapiro, J. H. / On convergence to the Denjoy-wolff point. In: Illinois Journal of Mathematics. 2005 ; Vol. 49, No. 2. pp. 405-430.
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