A model for planar compacta and rational Julia sets

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.