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 $Φ$ of $\mathbb{D}$ if and only if $Φ$ 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 $\mathbb{D}$.

Theorems & Definitions (35)

• Definition 1.1: Abel universal functions
• Theorem 2.1
• Theorem 2.2: Theorem 3 in Kaplan1954
• Theorem 2.3
• proof
• proof : Proof of Theorem \ref{['left_preserve']}
• Corollary 2.4
• Definition 2.5
• Definition 2.6
• Theorem 2.7