Frequently hypercyclic meromorphic curves with slow growth

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.

💡 Research Summary

The paper investigates the interplay between growth rates and frequent hypercyclicity for holomorphic maps from the complex plane into complex projective spaces. Starting from Birkhoff’s classical result on hypercyclic translations of entire functions, the authors move beyond linear operators to consider translation operators acting on spaces of holomorphic curves (H(\mathbb{C},\mathbb{P}^m)). The growth of such curves is measured by the Nevanlinna–Shimizu–Ahlfors characteristic function (T_h(r)). A fundamental lower bound (T_h(r) \ge C r) (or equivalently order (\rho_h \ge 1)) follows from the first main theorem of Nevanlinna theory when a curve is frequently hypercyclic for a non‑zero translation.

The first main result (Theorem 1.1) shows that for any prescribed countable set of directions (E\subset