Abstract
For sequences (φ n ) of eventually injective holomorphic self-maps of planar domains Ω, we present necessary and sufficient conditions for the existence of holomorphic functions f on whose orbits under the action of (φ n ) are dense in H (Ω). It is deduced that finitely connected, but non-simply connected domains never admit such universal functions. On the other hand, if we allow arbitrary sequences of holomorphic self-maps (φ n ), the situation changes dramatically.
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Grosse-Erdmann, KG., Mortini, R. Universal functions for composition operators with non-automorphic symbol. J Anal Math 107, 355–376 (2009). https://doi.org/10.1007/s11854-009-0013-4
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DOI: https://doi.org/10.1007/s11854-009-0013-4