Geodesic
On the existence of cut points of connected generalized Sierpiński carpets
Huo-Jun Ruan1 ; Yang Wang2 ; Jian-Ci Xiao1
1Zhejiang University, School of Mathematical Sciences
2The Hong Kong University of Science and Technology, Department of Mathematics
Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 229-254

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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.
DOI : 10.54330/afm.127049
Keywords: Generalized Sierpiński carpets, cut points, connectedness, Hata graphs

Huo-Jun Ruan  1   ; Yang Wang  2   ; Jian-Ci Xiao  1

1 Zhejiang University, School of Mathematical Sciences
2 The Hong Kong University of Science and Technology, Department of Mathematics 
Huo-Jun Ruan; Yang Wang; Jian-Ci Xiao. On the existence of cut points of connected generalized Sierpiński carpets. Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 229-254. doi: 10.54330/afm.127049
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