Generating all Ahlfors currents by a single entire curve

TL;DR

This paper constructs a single entire curve on certain complex manifolds that generates all Ahlfors currents, resolving a conjecture and advancing understanding of complex dynamics and approximation properties.

Contribution

It demonstrates that on manifolds with the Runge approximation property, one entire curve can generate all Ahlfors currents, confirming a longstanding conjecture.

Findings

Constructed an entire curve generating all Ahlfors currents.

Confirmed the conjecture of Sibony regarding Ahlfors currents.

Showed the importance of the Runge approximation property for such constructions.

Abstract

Let \(X\) be a compact complex manifold possessing the \emph{Runge approximation property on discs}, meaning that every holomorphic map from a closed disc into \(X\) is approximable by a global holomorphic map from \(\mathbb{C}\). We construct an entire curve \(F : \mathbb{C} \to X\) such that the associated family of concentric holomorphic discs \(\{F|_{\overline{\mathbb{D}}_r}\}_{r>0}\) generates all Ahlfors currents on \(X\), thereby settling a conjecture of Sibony.