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Bounded universal functions for sequences of holomorphic self-maps of the disk. (English) Zbl 1195.30069
Summary: We give several characterizations of those sequences of holomorphic self-maps $$\{\varphi_n \}_{n\geq 1}$$ 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 $$H^{\infty }$$ such that the orbit $$\{F\circ \varphi_n :n\in \mathbb N\}$$ 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 $$\varphi_n$$. As a consequence, we will see that if $$\varphi_n$$ is the $$n$$th iterate of a map $$\varphi$$ of $$\mathbb{D}$$ into $$\mathbb{D}$$, then $$\{\varphi_n \}_{n\geq 1}$$ admits a $$\mathcal{B}$$-universal function if and only if $$\varphi$$ 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.

##### MSC:
 30H05 Spaces of bounded analytic functions of one complex variable 30J10 Blaschke products
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