Abstract
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.
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