Cheap Complex Limit Cycles

TL;DR

This paper establishes a new criterion for holomorphic foliations on complex surfaces to possess infinitely many homologically independent complex limit cycles, especially when leaves are dense and a hyperbolic singularity exists.

Contribution

It introduces a sufficient condition linking dense leaves and hyperbolic singularities to the existence of infinitely many complex limit cycles.

Findings

Foliations with dense leaves and a hyperbolic singularity have infinitely many homologically independent limit cycles.

Provides a new criterion for the existence of complex limit cycles in holomorphic foliations.

Enhances understanding of the structure of limit cycles in complex dynamical systems.

Abstract

Consider a holomorphic foliation with singularities of a 2-dimensional complex manifold. In this article we prove a new sufficient condition for this foliation to have countably many homologically independent complex limit cycles. In particular, if all leaves of a foliation are dense in the phase space, and it has a complex hyperbolic singular point, then it has infinitely many homologically independent complex limit cycles.