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arXiv · 2209.03773

A model for planar compacta and rational Julia sets

Jun Luo·Yi Yang·Xiaoting Yao
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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.

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Jun Luo, Yi Yang, Xiaoting Yao. 2022-09-08. A model for planar compacta and rational Julia sets. https://arxiv.org/abs/2209.03773

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