Frequently hypercyclic meromorphic curves with slow growth

TL;DR

This paper constructs entire curves in projective spaces that are frequently hypercyclic along countably many directions with slow growth, and proves such behavior cannot occur for uncountably many directions under similar growth constraints.

Contribution

It introduces a novel construction of frequently hypercyclic entire curves with slow growth and establishes a fundamental dichotomy contrasting classical hypercyclicity phenomena.

Findings

Constructed entire curves with frequent hypercyclicity along countably many directions.

Proved impossibility of such curves for uncountably many directions under growth constraints.

Revealed a contrast with classical hypercyclicity where uncountable directions are possible.

Abstract

We construct entire curves in projective spaces that exhibit frequent hypercyclicity under translations along countably many prescribed directions while maintaining optimal slow growth rates. Furthermore, we establish a fundamental dichotomy by proving the impossibility of such curves simultaneously preserving frequent hypercyclicity for uncountably many directions under equivalent growth constraints. This result reveals a striking contrast with classical hypercyclicity phenomena, where entire functions can achieve hypercyclicity over some uncountable direction set without growth rate compromise. Our methodology is rooted in Nevanlinna theory and guided by the Oka principle, offering new insights into the relationship between dynamical properties and growth rates of entire curves in projective spaces.