Abstract
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 nth 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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Bayart, F., Gorkin, P., Grivaux, S. et al. Bounded universal functions for sequences of holomorphic self-maps of the disk . Ark Mat 47, 205–229 (2009). https://doi.org/10.1007/s11512-008-0083-z
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DOI: https://doi.org/10.1007/s11512-008-0083-z