Finitely Suslinian models for planar compacta with applications to Julia sets

TL;DR

This paper introduces a unique finitely Suslinian model for unshielded planar compacta, providing a framework to analyze Julia sets and branched coverings through monotone maps and quotient spaces.

Contribution

It establishes the existence and uniqueness of the finest finitely Suslinian model for unshielded planar compacta and extends this to Julia sets and branched coverings.

Findings

Existence of a unique monotone map to a finitely Suslinian quotient.

Extension of the model to branched covering maps and Julia sets.

Application to locally connected models of Julia sets.

Abstract

A compactum $X\subset \C$ is unshielded if it coincides with the boundary of the unbounded component of $\C\sm X$. Call a compactum $X$ finitely Suslinian if every collection of pairwise disjoint subcontinua of $X$ whose diameters are bounded away from zero is finite. We show that any unshielded planar compactum $X$ admits a topologically unique monotone map $m_X:X \to X_{FS}$ onto a finitely Suslinian quotient such that any monotone map of $X$ onto a finitely Suslinian quotient factors through $m_X$. We call the pair $(X_{FS},m_X)$ (or, more loosely, $X_{FS}$) the finest finitely Suslinian model of $X$. If $f:\C\to \C$ is a branched covering map and $X \subset \C$ is a fully invariant compactum, then the appropriate extension $M_X$ of $m_X$ monotonically semiconjugates $f$ to a branched covering map $g:\C\to \C$ which serves as a model for $f$. If $f$ is a polynomial and $J_f$ is its Julia set, we show that $m_X$ (or $M_X$) can be defined on each component $Z$ of $J_f$ individually as the finest monotone map of $Z$ onto a locally connected continuum.