Buried points of plane continua
David Lipham, Jan van Mill, Murat Tuncali, Ed Tymchatyn, Kirsten Valkenburg
TL;DR
This work investigates buried points on plane continua and proves that if the buried set bur(X) is totally disconnected, then it is zero-dimensional at all but countably many points under mild boundary connectivity assumptions; equivalently, bur(X) is either zero-dimensional or weakly 1-dimensional, and it cannot be almost zero-dimensional unless it is zero-dimensional. The authors introduce a local contraction tool F(y) and leverage Suslinian properties to bound the dimensionality of bur(X), sharpening prior examples (e.g., van Mill–Tuncali) and showing optimality in the countable sense. They further demonstrate the existence of locally connected van Mill–Tuncali-type continua and provide constructions yielding bur(X) with prescribed dimensional behavior, including cases where bur(X) is totally disconnected but 1-dimensional on a countable set. The results have implications for the structure of endpoints in plane dendroids and raise questions about buried sets in Julia sets of rational maps.
Abstract
Sets on the boundary of a complementary component of a continuum in the plane have been of interest since the early 1920's. Curry and Mayer defined the buried points of a plane continuum to be the points in the continuum which were not on the boundary of any complementary component. Motivated by their investigations of Julia sets, they asked what happens if the set of buried points of a plane continuum is totally disconnected and non-empty. Curry, Mayer and Tymchatyn showed that in that case the continuum is Suslinian, i.e. it does not contain an uncountable collection of non-degenerate pairwise disjoint subcontinua. In an answer to a question of Curry et al, van Mill and Tuncali constructed a plane continuum whose buried point set was totally disconnected, non-empty and one-dimensional at each point of a countably infinite set. In this paper we show that the van Mill-Tuncali example was best possible in the sense that whenever the buried set is totally disconnected, then it is one-dimensional at each of at most countably many points. As a corollary we find that the buried set cannot be almost zero-dimensional unless it is zero-dimensional. We also construct locally connected van Mill-Tuncali type examples.