Bounded universal functions for sequences of holomorphic self-maps of the disk

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.