A model for planar compacta and rational Julia sets

A Peano compactum is a compact metric space with 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. For any atom d of K and any finite-to-one open map f of the plane onto itself, we show that the preimage of d under f has finitely many components each of which is also an atom of the preimage of K under f. Since all rational functions are finite-to-one open maps, our result extends earlier ones that are restricted to more limited cases, requiring that f be a polynomial and K completely invariant under f. We also determine the atoms for typical planar continua. In particular, we show that every quadratic Julia set J that contains a Cremer fixed point has a unique atom, the whole Julia set J.