Lipschitz equivalence of self-similar sets with touching structures

Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar sets have touching structures the problem of Lipschitz equivalence becomes much more challenging and intriguing at the same time. So far, all the known results only cover self-similar sets in R with no more than three branches. In this study we establish results for the Lipschitz equivalence of self-similar sets with touching structures in R with arbitrarily many branches. Key to our study is the introduction of a geometric condition for self-similar sets called substitutable.