Meromorphic functions with linearly distributed values and Julia sets of rational functions

TL;DR

This paper explores the geometric properties of meromorphic functions with linearly distributed values and applies these findings to holomorphic dynamics, specifically characterizing Julia sets of rational functions contained in circles.

Contribution

It establishes a link between the preimages of four-point sets and the geometric shape of Julia sets in rational dynamics.

Findings

Preimage of four-point sets under meromorphic functions lies on the real line.

Julia sets contained in smooth curves are actually circles.

Provides a new geometric criterion for Julia sets in rational functions.

Abstract

If the preimage of a four-point set under a meromorphic function belongs to the real line, then the image of the real line is contained in a circle in the Riemann sphere. We include an application of this result to holomorphic dynamics: if the Julia set of a rational function is contained in a smooth curve then it is contained in a circle.