# Recurrent Trajectories and Finite Critical Trajectories of Quadratic Differentials on the Riemann Sphere

## TL;DR

This paper investigates the existence and non-existence of finite critical and recurrent trajectories of quadratic differentials on the Riemann sphere, establishing criteria and connections between these trajectory types.

## Contribution

It introduces new criteria for the non-existence of recurrent trajectories, extending Jenkins' Three-pole Theorem and relating to the Level function of quadratic differentials.

## Key findings

- Criteria for non-existence of recurrent trajectories
- Connection between finite critical and recurrent trajectories
- Extension of Jenkins' Three-pole Theorem

## Abstract

In this paper, the focus will be on both the existence and non-existence respectively of finite critical trajectories and recurrent trajectories of a quadratic differential on the Riemann sphere. We show the connection between these two items. More precisely, we collect some criterions for the non-existence of recurrent trajectories. The first criterion is in the same vein of Jenkins Three-pole Theorem, while the second one is in relation with the so-called Level function of quadratic differentials.