Irreducible Julia sets of rational functions

TL;DR

This paper classifies certain Julia sets of rational functions, showing they are either simple arcs or indecomposable continua, and provides a topological model for their dynamics in specific cases.

Contribution

It extends the classification of Julia sets to rational functions with irreducible continua, introducing a topological model for their dynamics.

Findings

Julia sets of polynomial functions are either arcs or indecomposable continua.

For rational functions, a topological model describes dynamics when the Julia set is irreducible with empty interior indecomposables.

The work generalizes known results from polynomials to rational functions.

Abstract

We prove that a polynomial Julia set which is a finitely irreducible continuum is either an arc or an indecomposable continuum. For the more general case of rational functions, we give a topological model for the dynamics when the Julia set is an irreducible continuum and all indecomposable subcontinua have empty interior.