Radial departures and plane embeddings of arc-like continua

TL;DR

This paper addresses a classical problem in topology by developing a new criterion based on radial departures to determine when arc-like continua can be embedded in the plane with a specified accessible point.

Contribution

It introduces the concept of radial departure and provides a criterion for embedding arc-like continua with accessible points, partially solving a problem posed in 1972.

Findings

Established a criterion for embedding arc-like continua with accessible points.

Provided a partial affirmative answer to Nadler and Quinn's problem.

Developed the notion of radial departure for inverse systems.

Abstract

We study the problem of Nadler and Quinn from 1972, which asks whether, given an arc-like continuum $X$ and a point $x \in X$, there exists an embedding of $X$ in $\mathbb{R}^2$ for which $x$ is an accessible point. We develop the notion of a radial departure of a map $f \colon [-1,1] \to [-1,1]$, and establish a simple criterion in terms of the bonding maps in an inverse system on intervals to show that there is an embedding of the inverse limit for which a given point is accessible. Using this criterion, we give a partial affirmative answer to the problem of Nadler and Quinn, under some technical assumptions on the bonding maps of the inverse system.