Universal holomorphic maps with slow growth II. functional analysis methods

TL;DR

This paper uses hypercyclic operator theory to show that holomorphic maps with slow growth are abundant in a topological sense, and demonstrates the existence of dense sets of frequently hypercyclic maps in certain function spaces.

Contribution

It introduces the concept of abstract abundance of hypercyclic holomorphic maps and establishes the density of frequently hypercyclic maps in the space of holomorphic functions to Oka manifolds.

Findings

Holomorphic maps with slow growth are topologically abundant.

Dense subsets of frequently hypercyclic maps exist in the space of holomorphic functions.

New results even for the case when domain is one-dimensional and target is complex Euclidean space.

Abstract

By means of hypercyclic operator theory, we complement our previous results on hypercyclic holomorphic maps between complex Euclidean spaces having slow growth rates,by showing {\it abstract abundance} rather than {\it explicit existence}. Next, we establish that, in the space of holomorphic maps from $\mathbb{C}^n$ to any connected Oka manifold $Y$, equipped with the compact-open topology, there exists a {\em dense} subset consisting of common {\em frequently hypercyclic} elements for all nontrivial translation operators. To our knowledge, this is new even for $n=1$ and $Y=\mathbb{C}$.