Fibers and local connectedness of planar continua

TL;DR

This paper introduces fiber-based concepts to characterize non-locally connected planar continua, establishing a scale for non-local connectedness and linking it to local properties and Julia sets.

Contribution

It defines the modified fiber and scale of non-local connectedness, providing new tools to analyze and classify planar continua's local connectedness properties.

Findings

The modified fiber $F_x^*$ is a continuum when points cannot be separated by finite sets.

Local connectedness at a point is characterized by trivial fibers.

Constructs examples of continua with arbitrary non-local connectedness scale.

Abstract

We describe non-locally connected planar continua via the concepts of fiber and numerical scale. Given a continuum $X\subset\mathbb{C}$ and $x\in\partial X$, we show that the set of points $y\in \partial X$ that cannot be separated from $x$ by any finite set $C\subset \partial X$ is a continuum. This continuum is called the {\em modified fiber} $F_x^*$ of $X$ at $x$. If $x\in X^o$, we set $F^*_x=\{x\}$. For $x\in X$, we show that $F_x^*=\{x\}$ implies that $X$ is locally connected at $x$. We also give a concrete planar continuum $X$, which is locally connected at a point $x\in X$ while the fiber $F_x^*$ is not trivial. The scale $\ell^*(X)$ of non-local connectedness is then the least integer $p$ (or $\infty$ if such an integer does not exist) such that for each $x\in X$ there exist $k\le p+1$ subcontinua $$X=N_0\supset N_1\supset N_2\supset\cdots\supset N_{k}=\{x\}$$ such that $N_{i}$ is a fiber of $N_{i-1}$ for $1\le i\le k$. If $X\subset\mathbb{C}$ is an unshielded continuum or a continuum whose complement has finitely many components, we obtain that local connectedness of $X$ is equivalent to the statement $\ell^*(X)=0$. We discuss the relation of our concepts to the works of Schleicher (1999) and Kiwi (2004). We further define an equivalence relation $\sim$ based on the fibers and show that the quotient space $X/\sim$ is a locally connected continuum. For connected Julia sets of polynomials and more generally for unshielded continua, we obtain that every prime end impression is contained in a fiber. Finally, we apply our results to examples from the literature and construct for each $n\ge1$ concrete examples of path connected continua $X_n$ with $\ell^*(X_n)=n$.