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

Frédéric Bayart Université Bordeaux 1, 351 Cours de la Libération, Talence, France Pamela Gorkin Department of Mathematics, Bucknell University Sophie Grivaux Laboratoire Paul Painlevé, UMR 8524, Université des Sciences et Technologies de Lille Raymond Mortini Département de Mathématiques, Université Paul Verlaine

TBD mathscidoc:1701.333152

Arkiv for Matematik, 47, (2), 205-229, 2007.7
We give several characterizations of those sequences of holomorphic self-maps {φ_{$n$}}_{$n$≥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 {$F$∘φ_{$n$}:$n$∈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 φ_{$n$}. As a consequence we will see that if φ_{$n$}is the$n$th iterate of a map φ of $\mathbb{D}$ into $\mathbb{D}$ , then {φ_{$n$}}_{$n$≥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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@inproceedings{frédéric2007bounded,
  title={Bounded universal functions for sequences of holomorphic self-maps of the disk},
  author={Frédéric Bayart, Pamela Gorkin, Sophie Grivaux, and Raymond Mortini},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203625365560961},
  booktitle={Arkiv for Matematik},
  volume={47},
  number={2},
  pages={205-229},
  year={2007},
}
Frédéric Bayart, Pamela Gorkin, Sophie Grivaux, and Raymond Mortini. Bounded universal functions for sequences of holomorphic self-maps of the disk. 2007. Vol. 47. In Arkiv for Matematik. pp.205-229. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203625365560961.
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