Many

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 ≥ 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 ≥ 1 a connected GSC with exactly m cut points and demonstrate that when m ≥ 2, such a GSC must be of the so-called fragile type.