The study of Lipschitz equivalence of fractals is a very active topic in recent years. Most of the studies in literature concern totally disconnected fractals. In this paper, using neighbor automata of self-similar sets and a transducer, we construct a bi-Lipschitz map between two fractal squares which are not totally disconnected. This is the first nontrivial construction of this type. We also show that this map is measure-preserving.

中文翻译：

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自相似集和有限状态自动机的 LIPSCHITZ 等价性

分形的 Lipschitz 等价性研究是近年来非常活跃的课题。文献中的大多数研究都涉及完全不连贯的分形。在本文中，使用自相似集的邻域自动机和一个换能器，我们在两个不完全断开的分形正方形之间构建了一个双李普希茨图。这是这种类型的第一个重要构造。我们还表明，这张地图是保持测量的。