Topological classification of self-similar subsets of the line

Classify self-similar sets in the real line into a finite collection of topological types, or determine whether no such rigid finite classification is possible.

Background

For achievement sets of convergent positive series, the paper invokes the established trichotomy that every such set is a finite union of intervals, a Cantor set, or a Cantorval. The family studied in the paper fits within this achievement-set framework, enabling a complete classification for its particular parameterisation.

The authors note that not every self-similar subset of the line is an achievement set. They therefore leave unresolved whether a comparable finite topological classification exists for arbitrary self-similar sets in the line.