Radial departures and plane embeddings of arc-like continua

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 $f:[-1,1]\to [-1,1]$, 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.

Some typos fixed. Numeration of theorems and lemmas is changed