Non-degenerate locally connected models for plane continua and Julia sets

TL;DR

This paper investigates conditions under which unshielded plane continua, including Julia sets, have meaningful locally connected models, providing insights into their structure and decomposition.

Contribution

It establishes sufficient conditions for the existence of non-degenerate locally connected models of plane continua, especially Julia sets, based on subcontinua properties.

Findings

Sufficient conditions for non-degenerate models are identified.

Application of results to polynomial Julia sets.

Provides criteria for when models are singleton or non-singleton.

Abstract

Suppose that a $X$ is an \emph{unshielded} plane continuum (i.e., $X$ coincides with the boundary of the unbounded complementary component of $X$). Then there exists a \emph{finest monotone} map $m:X\to L$, where $L$ is a locally connected continuum (i.e., $m^{-1}(y)$ is connected for each $y\in L$, and any monotone map $\varphi:X\to L'$ onto a locally connected continuum is a composition $\varphi=\varphi'\circ m$ where $\varphi':L\to L'$ is monotone). Such finest locally connected model $L$ of $X$ is easier to understand because $L$ is locally connected (in particular it can be described by a picture) and represents the finest but still understandable decomposition of $X$ into possibly complicated but pairwise disjoint \emph{fibers} (point-preimages) of $m$. However, in some cases (i.e., in case $X$ is indecomposable) $L$ is a singleton. In this paper we provide sufficient conditions for the existence of a non-degenerate model depending on the existence of certain subcontinua of $X$ and apply these results to the connected Julia sets of polynomials.