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Abstract. For sequences (φn) of eventually injective holomorphic self-maps of planar domains Ω we present necessary and sufficient conditions for the existence of holomorphic functions f on Ω whose orbits under the action of (φn) 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, when allowing arbitrary sequences of holomorphic selfmaps (φn), then we show that the situation changes dramatically.

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T. Hosokawa, K. Izuchi and D. Zheng recently introduced the concept of asymptotic interpolating sequences (of type 1) in the unit disk for H ∞ (D). It is shown that these sequences coincide with sequences that are interpolating for the algebra QA. Also a characterization is given of the interpolating sequences of type 1 for H ∞ (D), and asymptotic interpolating sequences in the spectrum of H ∞ (D) are studied. The existence of asymptotic interpolating sequences of type 1 for H ∞ (Ω) on arbitrary domains is verified. It is shown that any asymptotic interpolating sequence in a uniform algebra eventually is interpolating.

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