A model for planar compacta and rational Julia sets

A Peano compactum means 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 $\hat{C}$, 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 $f:\hat{C}\to \hat{C}$ and any atom d of $f^{-1}\left(K\right)$, the image $f\left(d\right)$ is an atom of K. Since rational functions are branched coverings, our result extends earlier ones that are restricted to more limited cases, requiring that f be a polynomial with $f^{-1}\left(K\right)=K$.