Abstract
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.
Citation
Frédéric Bayart. Pamela Gorkin. Sophie Grivaux. Raymond Mortini. "Bounded universal functions for sequences of holomorphic self-maps of the disk." Ark. Mat. 47 (2) 205 - 229, October 2009. https://doi.org/10.1007/s11512-008-0083-z
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