Invariance of Abel universality under composition and applications

Abstract

A holomorphic function $f$ on the unit disc $\mathbb{D}$ belongs to the class $\mathcal{U}_A (\mathbb{D})$ of Abel universal functions if the family $\{f_r: 0\leq r<1\}$ of its dilates $f_r(z):=f(rz)$ is dense in the Banach space of all continuous functions on $K$, endowed with the supremum norm, for any proper compact subset $K$ of the unit circle. We prove that this property is invariant under composition from the left with any non-constant entire function. As an application, we show that $\mathcal{U}_A (\mathbb{D})$ is strongly-algebrable. Furthermore, we prove that Abel universality is invariant under composition from the right with an automorphism $\Phi$ of $\mathbb{D}$ if and only if $\Phi$ a rotation. On the other hand, we establish the existence of a subset of $\mathcal{U}_A (\mathbb{D})$ which is residual in the space of holomorphic functions on $\mathbb{D}$ and is invariant under composition from the right with any automorphism of $\mathbb{D}$.