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

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 $C$ of composants of $K$, there exists a plane embedding of $K$ in which every point in the union of the composants in $C$ is accessible. We also exhibit new embeddings of the Knaster bucket-handle 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 $K$.