Fractal necklaces with no cut points

TL;DR

This paper studies a class of connected fractal sets called fractal necklaces generated by IFSs, establishing conditions under which these sets have no cut points, with implications for topology and self-similarity.

Contribution

It introduces two subclasses of fractal necklaces and proves they have no cut points, also analyzing the topological properties of stable self-similar necklaces.

Findings

Fractal necklaces in certain subclasses have no cut points.

Stable self-similar necklaces in R^2 lack cut points.

Self-affine necklaces can have cut points, unlike self-similar ones.

Abstract

The fractal necklaces in R^d (d>1) introduced in this paper are a class of connected fractal sets generated by the so-called necklace IFSs, for which a lot of basic topology questions are interesting. We give two subclasses of fractal necklaces and prove that every necklace in these two classes has no cut points. Also, we prove that every stable self-similar necklace in R^2 has no cut points, whilst an analog for self-affine necklaces is false.