Planar embeddings of Minc's continuum and generalizations

TL;DR

This paper demonstrates that for certain piecewise monotone, post-critically finite functions, their inverse limit spaces can be embedded in the plane with all points accessible, including Minc's continuum, using thin embeddings.

Contribution

It provides a method to embed inverse limit spaces of specific functions into the plane with accessibility for all points, extending to Minc's continuum.

Findings

All points in the inverse limit space can be made accessible in the planar embedding.

Embeddings are thin, allowing coverage by small chains of connected open sets.

The method applies to Minc's continuum, answering a specific open question.

Abstract

We show that if $f\colon I\to I$ is piecewise monotone, post-critically finite, and locally eventually onto, then for every point $x\in X=\underleftarrow{\lim}(I,f)$ there exists a planar embedding of $X$ such that $x$ is accessible. In particular, every point $x$ in Minc's continuum $X_M$ from [Question 19 p. 335 in Continuum theory : proceedings of the special session in honor of Professor Sam B. Nadler, Jr.'s 60th birthday, Lecture notes in pure and applied mathematics; v. 230, New York: Marcel Dekker.] can be embedded accessibly. All constructed embeddings are thin, i.e. can be covered by an arbitrary small chain of open sets which are connected in the plane.