Accessibility of countable sets in plane embeddings of arc-like continua

TL;DR

This paper investigates conditions for embedding arc-like continua in the plane with accessible points, demonstrating the existence of such embeddings for specific inverse systems and continua like the Knaster continuum, including applications to attractors of plane homeomorphisms.

Contribution

It establishes new conditions for accessible embeddings of arc-like continua in the plane, including for Knaster continua, and constructs embeddings that are attractors of plane homeomorphisms.

Findings

Existence of accessible embeddings for inverse limits of arcs under certain conditions.

Construction of plane embeddings of Knaster continua with accessible points in specified composants.

New embeddings of the Knaster buckethandle continuum as attractors conjugate to shift maps.

Abstract

We consider the problem of finding embeddings of arc-like continua in the plane for which each point in a given subset is accessible. We establish that, under certain conditions on an inverse system of arcs, there exists a plane embedding of the inverse limit for which each point of a given countable set is accessible. As an application, we show that for any Knaster continuum $K$, and any countable collection $\mathcal{C}$ of composants of $K$, there exists a plane embedding of $K$ in which every point in the union of the composants in $\mathcal{C}$ is accessible. We also exhibit new embeddings of the Knaster buckethandle continuum $K$ in the plane which are attractors of plane homeomorphisms, and for which the restriction of the plane homeomorphism to the attractor is conjugate to a power of the standard shift map on $K$.