In this paper, we present an effective method to characterize completely when a disconnected fractal square has only finitely many connected components. Our method is to establish some graph structures on fractal squares to reveal the evolution of the connectedness during their geometric iterated construction. We also prove that every fractal square contains either finitely or uncountably many connected components. A few examples, including the construction of fractal squares with exactly m ⩾ 2 connected components, are added in addition.

中文翻译：

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具有有限多个连通分量的分形平方这项工作得到了 NSFC 资助 11771391 的部分支持。

在本文中，我们提出了一种有效的方法来完全表征当不连续的分形正方形只有有限多个连通分量时。我们的方法是在分形正方形上建立一些图结构，以揭示其几何迭代构造过程中连通性的演变。我们还证明了每个分形正方形都包含有限或不可数的许多连通分量。此外还添加了一些示例，包括构造具有恰好m ⩾ 2 个连通分量的分形正方形。