2022/09/08 by Jun Luo, Yi Yang, Luo, Jun +3
Mathematics · #2020: 37B45 #37F10 #54D05 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2209.03773
openalex publication_date 2022/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.