Existence of rational Julia sets with totally disconnected but non-zero-dimensional buried sets

Determine whether there exists a rational function R on the Riemann sphere such that the set of buried points bur(J(R)) of its Julia set J(R)—the points of J(R) not lying on the boundary of any complementary component—is totally disconnected yet not zero-dimensional.

Background

The paper investigates the topology of buried points in plane continua, focusing on constraints when the buried set is totally disconnected. The main result (Theorem A) shows that in a Suslinian plane continuum any totally disconnected Borel subset is zero-dimensional at all but countably many points; as a consequence (Corollary B), for plane continua whose complementary component boundaries are locally connected, a totally disconnected buried set is zero-dimensional at all but countably many points.

While the authors construct locally connected plane continua with buried sets that are totally disconnected but not zero-dimensional, it remains unresolved whether such a phenomenon can occur for Julia sets of rational functions. This bridges the paper’s topological results with a central open question in complex dynamics, specifically concerning the dimensional nature of buried points within Julia sets.