Planar embeddings of chainable continua

A continuum is a compact and connected metric space, and it is chainable if it can be covered by an arbitrary small chain, or equivalently if it can be represented as an inverse limit on intervals.  It is well-known that every chainable continuum can be embedded in the plane (Bing 1951,  Anderson and Choquet 1959), but it is still not known how to describe possibly different (non-equivalent)  planar embeddings of a chainable continuum.

For example, one might wonder which points in a planar  representation are accessible, and which are not. A point p in a planar continuum X is called accessible  (from the complement of X) if there is an arc A in the plane which intersects X only in p. Nadler and Quinn  asked in 1972 whether for every chainable continuum X and a point p contained in X, there exists a planar embedding of X such that p is accessible. This question is still open. We discuss some special cases in which we have a positive answer (A, Bruin, Cinc 2018), and show that for every interval map f which is piecewise monotone, post-critically finite, and locally eventually onto, every point of lim(I,f) can be embedded accessible (A 2020). In particular, every point of Minc's continuum can be embedded accessible. Minc's continuum was  introduced in 2001, and was considered to be a likely counterexample to the Nadler-Quinn question.

Time permitting, we will discuss some ideas for further generalization, which are part of current research with L. Hoehn.