On the existence of cut points of connected generalized Sierpiński carpets

Abstract

In a previous work joint with Dai and Luo, we show that a connected generalized Sierpiński carpet (or shortly a GSC) has cut points if and only if the associated $n$-th Hata graph has a long tail for all $n\ge 2$. In this paper, we extend the above result by showing that it suffices to check a finite number of those graphs to reach a conclusion. This criterion provides a truly "algorithmic" solution to the cut point problem of connected GSCs. We also construct for each $m\ge 1$ a connected GSC with exactly $m$ cut points and demonstrate that when $m\ge 2$, such a GSC must be of the so-called non-fragile type.