Connectedness and local cut points of generalized Sierpinski carpets

TL;DR

This paper characterizes when generalized Sierpinski carpets are homeomorphic to the standard Sierpinski carpet by analyzing connectedness, cut points, and local cut points, and extends these results to Baranski carpets and sponges.

Contribution

It provides new criteria for determining connectedness and local cut points in GSCs, enabling complete classification relative to the standard Sierpinski carpet.

Findings

Connectedness can be determined after two iterations of the initial pattern.

Criteria for the existence of cut points in GSCs.

Criteria for the existence of local cut points in GSCs.

Abstract

We investigate a homeomorphism problem on a class of self-similar sets called generalized Sierpinski carpets (or shortly GSCs). It follows from two well-known results by Hata and Whyburn that a connected GSC is homeomorphic to the standard Sierpinski carpet if and only if it has no local cut points. On the one hand, we show that to determine whether a given GSC is connected, it suffices to iterate the initial pattern twice. On the other hand, we obtain two criteria: (1) for a connected GSC to have cut points, (2) for a connected GSC with no cut points to have local cut points. With these two criteria, we characterize all GSCs that are homeomorphic to the standard Sierpinski carpet. Our results on cut points and local cut points hold for Baranski carpets, too. Moreover, we extend the connectedness result to Baranski sponges. Thus, we also characterize when a Baranski carpet is homeomorphic to the standard GSC.