Abel universal series

Abstract

Given a sequence = (r n) n $\in$ [0, 1) tending to 1, we consider the set U A (D,) of Abel universal series consisting of holomorphic functions f in the open unit disc D such that for any compact set K included in the unit circle T, different from T, the set {z $\rightarrow$ f (r n $\bullet$)| K : n $\in$ N} is dense in the space C(K) of continuous functions on K. It is known that the set U A (D,) is residual in H(D). We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at 0 are dense in C(K) for any compact set K $\subset$ T different from T. Moreover we prove that the class of Abel universal series is not invariant under the action of the differentiation operator. Finally an Abel universal series can be viewed as a universal vector of the sequence of dilation operators T n : f $\rightarrow$ f (r n $\bullet$) acting on H(D). Thus we study the dynamical properties of (T n) n such as the multi-universality and the (common) frequent universality. All the proofs are constructive.