On invariant line fields of rational functions

TL;DR

This paper investigates the structure of invariant line fields on the Julia set of rational functions, showing they are determined by a finite set of canonical holomorphic line fields under certain conditions.

Contribution

It introduces a framework to classify invariant line fields on Julia sets, linking them to a finite set of canonical holomorphic line fields.

Findings

Invariant line fields are determined by a finite list of canonical holomorphic line fields.

Under certain conditions, all measurable invariant line fields are classified.

The approach extends previous methods to a broader class of rational functions.

Abstract

Adopting the approach of [7] we study rational function carrying invariant line fields on the Julia set. In particular, we show that under certain weak conditions all possible measurable invariant line fields of a rational function on its Julia set are determined, in a precise sense, by a finite list of canonically defined holomorphic line fields.