Lipschitz equivalence of self-similar sets and hyperbolic boundaries II

Abstract

In [13], two of the authors gave a study of Lipschitz equivalence of self-similar sets through the augmented trees, a class of hyperbolic graphs introduced by Kaimanovich in [9] and developed by Lau and Wang [10]. In this paper, we continue such investigation. We remove a major assumption in the main theorem in [13] by using a new notion of quasi-rearrangeable matrix, and show that the hyperbolic boundary of any simple augmented tree is Lipschitz equivalent to a Cantor-type set. We then apply this result to consider the Lipschitz equivalence of certain totally disconnected self-similar sets as well as their unions.