On the existence of cut points of connected generalized Sierpinski carpets

TL;DR

This paper provides an algorithmic criterion to determine the existence of cut points in connected generalized Sierpinski carpets by analyzing a finite set of associated graphs, extending previous infinite-graph conditions.

Contribution

It extends prior results by reducing the cut point criterion to finitely many graphs, enabling an algorithmic approach, and constructs GSCs with prescribed numbers of cut points.

Findings

A finite check suffices to determine cut points in connected GSCs.

Constructed GSCs with exactly m cut points for any m ≥ 1.

Connected GSCs with multiple cut points are of fragile type.

Abstract

In a previous work joint with Dai and Luo, we show that a connected generalized Sierpi\'nski carpet (or shortly a GSC) has cut points if and only if the associated $n$-th Hata graph has a long tail for all $n\geq 2$. 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 $m\geq 1$ a connected GSC with exactly $m$ cut points and demonstrate that when $m\geq 2$, such a GSC must be of the so-called fragile type.