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

We give several characterizations of those sequences of holomorphic self-maps {phi(n)}(n >= 1) of the unit disk for which there exists a function F in the unit ball B={f is an element of H(infinity): parallel to f parallel to(infinity)<= 1} of H(infinity) such that the orbit {F circle phi(n): n is an element of N} is locally uniformly dense in B. Such a function F is said to be a B-universal function. One of our conditions is stated in terms of the hyperbolic derivatives of the functions fn. As a consequence we will see that if phi(n) is the nth iterate of a map phi of D into D, then {phi(n)}(n >= 1) admits a B-universal function if and only if phi is a parabolic or hyperbolic automorphism of D. We show that whenever there exists a B-universal function, then this function can be chosen to be a Blaschke product. Further, if there is a B-universal function, we show that there exist uniformly closed subspaces consisting entirely of universal functions.