Universal holomorphic maps with slow growth II. functional analysis methods (2310.06561v3)

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 $Y=\mathbb{C}$.