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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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