A Core Decomposition of Compact Sets in the Plane

TL;DR

This paper introduces a unique core decomposition for compact sets in the plane, leading to a Peano model that is topologically determined and applicable to complex dynamics, with some sets lacking such a decomposition.

Contribution

It defines the core decomposition with Peano quotient for compact sets in the plane and shows its topological invariance and relation to existing models in complex dynamics.

Findings

Existence of a finest upper semi-continuous decomposition into Peano continua.

The Peano model is topologically determined and embedding-independent.

Counterexample of a compact set without a Peano core decomposition.

Abstract

A Peano continuum means a locally connected continuum. A compact metric space is called a \emph{Peano compactum} if all its components are Peano continua and if for any constant $C>0$ all but finitely many of its 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 Peano compactum. We prove that one of these decompositions is finer than all the others and call it the \emph{core decomposition of $K$ with Peano quotient}. This core decomposition gives rise to a metrizable quotient space, called the Peano model of $K$, which is shown to be determined by the topology of $K$ and hence independent of the embedding of $K$ into $\mathbb{C}$. We also construct a concrete continuum $K\subset\mathbb{R}^3$ such that the core decomposition of $K$ with Peano quotient does not exist. For specific choices of $K\subset\mathbb{C}$, the above mentioned core decomposition coincides with two models obtained recently, namely the locally connected model for unshielded planar continua (like connected Julia sets of polynomials) and the finitely Suslinian model for unshielded planar compact sets (like polynomial Julia sets that may not be connected). The study of such a core decomposition provides partial answers to several questions posed by Curry in 2010. These questions are motivated by other works, including those by Curry and his coauthors, that aim at understanding the dynamics of a rational map $f: \hat{\mathbb{C}}\rightarrow\hat{\mathbb{C}}$ restricted to its Julia set.