A Core Decomposition of Compact Sets in the Plane

Benoit Loridant, Jun Luo, Yi Yang

Published 2017-12-18Version 1

A compact metric space is called a \emph{generalized Peano space} if all its components are locally connected and if for any constant $C>0$ all but finitely many of the components are of diameter less than $C$. Given a compact set $K\subset\mathbb{C}$, there usually exist several upper semi-continuous decompositions of $K$ into subcontinua such that the quotient space, equipped with the quotient topology, is a generalized Peano space. We show that one of these decompositions is finer than all the others and call it the \emph{core decomposition of $K$ with Peano quotient}. For specific choices of $K$, this core decomposition coincides with two models obtained recently, namely the locally connected models for unshielded planar continua (like connected Julia sets of polynomials) and the finitely Suslinian models for unshielded planar compact sets (like disconnected Julia sets of polynomials). We further answer several questions posed by Curry in 2010. In particular, we can exclude the existence of a rational function whose Julia set is connected and does not have a finest locally connected model.