Topological Rigidity of good fractal necklaces

TL;DR

This paper studies the topological properties of good fractal necklaces, proving their uniqueness, rigidity of homeomorphisms, and a co-Hopfian property, thereby advancing understanding of their structural symmetries.

Contribution

It introduces extremal 2-cuts for good fractal necklaces and proves their implications for uniqueness and rigidity of the necklaces' homeomorphisms.

Findings

Every good necklace has a unique associated IFS.

The group of self-homeomorphisms of a good necklace is countable.

Good necklaces exhibit a weaker co-Hopfian property.

Abstract

We introduce and characterize extremal 2-cuts for good fractal necklaces. Using the characterization and the related topological properties of extremal 2-cuts, we prove that every good necklace has a unique necklace IFS in a certain sense. Also, we prove that two good necklaces admit only rigid homeomorphisms and thus the group of self-homeomorphisms of a good necklace is countable. In addition, a certain weaker co-Hopfian property of good necklaces is also obtained.