A model for planar compacta and rational Julia sets

TL;DR

This paper introduces a new decomposition method for compacta in the extended complex plane, showing how preimages of atoms under branched coverings, including rational functions, behave in terms of their atomic structure.

Contribution

It extends previous results by demonstrating that preimages of atoms under rational functions have finitely many atomic components, generalizing earlier polynomial-specific findings.

Findings

Preimages of atoms under branched coverings have finitely many atomic components.

The decomposition applies to any branched covering, including rational functions.

The results generalize previous polynomial-specific theorems.

Abstract

A Peano compactum is a compact metric space having locally connected components such that at most finitely many of them are of diameter greater than any fixed number C>0. Given a compactum K in the extended complex plane, it is known that there is a finest upper semi-continuous decomposition of K into subcontinua such that the resulting quotient space is a Peano compactum. We call this decomposition the core decomposition of K with Peano quotient and its elements atoms of K. We show that for any branched covering f of the extended complex plane onto itself and for any atom d of K, the preimage of d under f has finitely many components each of which is an atom of the preimage of K under f. Since rational functions are branched coverings, our result extends earlier ones that are restricted to more limited cases, requiring that f be a polynomial and K completely invariant under f.