Arkiv för Matematik

  • Ark. Mat.
  • Volume 47, Number 2 (2009), 205-229.

Bounded universal functions for sequences of holomorphic self-maps of the disk

Frédéric Bayart, Pamela Gorkin, Sophie Grivaux, and Raymond Mortini

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We give several characterizations of those sequences of holomorphic self-maps {φn}n≥1 of the unit disk for which there exists a function F in the unit ball $\mathcal{B}=\{f\in H^{\infty}: \|f\|_\infty\leq1\}$ of H such that the orbit {F∘φn: n∈ℕ} is locally uniformly dense in $\mathcal{B}$. Such a function F is said to be a $\mathcal{B}$-universal function. One of our conditions is stated in terms of the hyperbolic derivatives of the functions φn. As a consequence we will see that if φn is the nth iterate of a map φ of $\mathbb{D}$ into $\mathbb{D}$, then {φn}n≥1 admits a $\mathcal{B}$-universal function if and only if φ is a parabolic or hyperbolic automorphism of $\mathbb{D}$. We show that whenever there exists a $\mathcal{B}$-universal function, then this function can be chosen to be a Blaschke product. Further, if there is a $\mathcal{B}$-universal function, we show that there exist uniformly closed subspaces consisting entirely of universal functions.

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Ark. Mat., Volume 47, Number 2 (2009), 205-229.

Received: 27 July 2007
First available in Project Euclid: 31 January 2017

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2008 © Institut Mittag-Leffler


Bayart, Frédéric; Gorkin, Pamela; Grivaux, Sophie; Mortini, Raymond. Bounded universal functions for sequences of holomorphic self-maps of the disk. Ark. Mat. 47 (2009), no. 2, 205--229. doi:10.1007/s11512-008-0083-z.

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