arXiv subjects
Publications and source records attributed to Andrea Ammerlaan.
We show that if $X$ is an arc-like continuum, then any continuum which is the union of $X$ and a ray $R$ such that $X \cap R = \emptyset$ and $\overline{R} \setminus R \subseteq X$ can be embedded in the plane $\mathbb{R}^2$. Further, we prove that any compactification of a line with remainder $X$ is also embeddable in $\mathbb{R}^2$ -- answering a question of Sam B. Nadler from 1972.
We show that if $X$ is an arc-like continuum, then for every point $x \in X$ there is a plane embedding of $X$ in which $x$ is an accessible point. This answers a question posed by Nadler in 1972, which has become known as the Nadler-Quinn problem in continuum theory. Towards this end, we develop the theories of truncations and contour factorizations of interval maps. As a corollary, we answer a question of Mayer from 1982 about inequivalent plane embeddings of indecomposable arc-like continua.
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.
We show that if $X$ is an arc-like continuum which can be represented as an inverse limit of a simplicial inverse system on arcs, then for every point $x \in X$ there is a plane embedding of $X$ in which $x$ is accessible. This answers a special case of the Nadler-Quinn question from 1972.