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Frequently hypercyclic meromorphic curves with slow growth

Zhangchi Chen, Bin Guo, Song-Yan Xie

TL;DR

This work addresses the minimal growth requirements for frequent hypercyclicity of entire curves h: C → P^m under translations along a prescribed set of directions. It fuses Nevanlinna theory with Oka-principle insights to construct a slowly growing universal meromorphic curve that is frequently hypercyclic for a countable set of directions, and proves a density-based obstruction that prevents such optimal slow growth for uncountably many directions. The authors introduce a novel density-preserving lemma and a scaling technique to achieve T_h(r) ≤ ε r, providing a precise dichotomy between countable versus uncountable directional families. The results deepen the understanding of how dynamical properties of holomorphic curves interact with growth constraints in projective spaces, with potential applications to density-sensitive constructions in complex geometry.

Abstract

We construct entire curves in projective spaces that are simultaneously frequently hypercyclic for translations along countably many specified directions, while preserving optimal slow growth rates. Moreover, we demonstrate that there do not exist entire curves that are frequently hypercyclic for translations along uncountably many directions while still exhibiting optimal slow growth. This result contrasts with the phenomenon observed for hypercyclic entire functions, which can be hypercyclic for translations along uncountably many directions while maintaining slow growth. Our approach is fundamentally grounded in Nevanlinna theory, and the construction is heuristically guided by the Oka principle. This work provides a new perspective on the intricate interplay between the dynamical properties and growth rates of entire curves in projective spaces.

Frequently hypercyclic meromorphic curves with slow growth

TL;DR

This work addresses the minimal growth requirements for frequent hypercyclicity of entire curves h: C → P^m under translations along a prescribed set of directions. It fuses Nevanlinna theory with Oka-principle insights to construct a slowly growing universal meromorphic curve that is frequently hypercyclic for a countable set of directions, and proves a density-based obstruction that prevents such optimal slow growth for uncountably many directions. The authors introduce a novel density-preserving lemma and a scaling technique to achieve T_h(r) ≤ ε r, providing a precise dichotomy between countable versus uncountable directional families. The results deepen the understanding of how dynamical properties of holomorphic curves interact with growth constraints in projective spaces, with potential applications to density-sensitive constructions in complex geometry.

Abstract

We construct entire curves in projective spaces that are simultaneously frequently hypercyclic for translations along countably many specified directions, while preserving optimal slow growth rates. Moreover, we demonstrate that there do not exist entire curves that are frequently hypercyclic for translations along uncountably many directions while still exhibiting optimal slow growth. This result contrasts with the phenomenon observed for hypercyclic entire functions, which can be hypercyclic for translations along uncountably many directions while maintaining slow growth. Our approach is fundamentally grounded in Nevanlinna theory, and the construction is heuristically guided by the Oka principle. This work provides a new perspective on the intricate interplay between the dynamical properties and growth rates of entire curves in projective spaces.
Paper Structure (7 sections, 14 theorems, 102 equations)

This paper contains 7 sections, 14 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Nevanlinna Theory
  3. Preparations
  4. A Key Lemma
  5. Some Estimates
  6. Model Curves
  7. Proof of Theorem \ref{['main-thm-c']}

Key Result

Theorem 1.1

For any $m \geqslant 1$, for any continuous increasing function $\phi: [1, +\infty) \to (0, +\infty)$ tending to infinity, and for any countable set $E \subset [0, 2\pi)$, there exists a universal entire curve $h: \mathbb{C} \rightarrow \mathbb{P}^m$ with slow growth rate Moreover, $h$ is simultaneously hypercyclic with respect to every translation $\mathsf{T}_a$, where $a \in \mathbb{C} \setminu

Theorems & Definitions (28)

  • Theorem 1.1: Chen-Huynh-Xie-2023--Theorem 1.3
  • Theorem 1.2: Guo-Xie-2--Theorem D
  • Remark 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Proposition 2.1
  • proof
  • proof : Proof of Proposition \ref{['countable fhc direction']}
  • Lemma 3.1: Grosse-Erdmann-Manguillot-2011--Lemma 9.5
  • Proposition 3.2
  • ...and 18 more