Planar embeddings of chainable continua

Abstract

We prove that for a chainable continuum $X$ and every non-zigzag $x\in X$ there exists a planar embedding $\varphi :X\to \varphi (X)\subset {\mathbb{R}}^2$ such that $\varphi (x)$ is accessible, partially answering the question of Nadler and Quinn from 1972. Two embeddings $\varphi ,\psi :X\to {\mathbb{R}}^2$ are called strongly equivalent if $\varphi \circ {\psi }^{-1}:\psi (X)\to \varphi (X)$ can be extended to a homeomorphism of ${\mathbb{R}}^2$. We also prove that every indecomposable chainable continuum can be embedded in the plane in uncountably many strongly non-equivalent ways.

Corrected some typos and the proof of Theorem 8.5