Topological Rigidity of good fractal necklaces

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.