Topological invariants and Lipschitz equivalence of fractal squares

Fractal sets typically have very complex geometric structures, and a fundamental problem in fractal geometry is to characterize how “similar” different fractal sets are. The Lipschitz equivalence of fractal sets is often used to classify fractal sets that are geometrically similar. Interesting links between Lipschitz equivalence and algebraic properties of contraction ratios for self-similar sets have been uncovered and widely analyzed. However, with the exception of very few papers, the study of Lipschitz equivalence has largely focused on totally disconnected self-similar sets. For connected self-similar sets this problem becomes rather challenging, even for well known fractal models such as fractal squares. In this paper, we introduce geometric and topological methods to study the Lipschitz equivalence of connected fractal squares. In particular we completely characterize the Lipschitz equivalence of fractal squares of order 3 in which one or two squares are removed. We also discuss the Lipschitz equivalence of fractal squares of more general orders. Our paper is the first study of Lipschitz equivalence for nontrivial connected self-similar sets, and it raises also some interesting questions for the more general setting.