Let C be the triadic Cantor set. We characterize the all real number α such that the intersection C∩(C+α) is a self-similar set, and investigate the form and structure of the all iterated function systems which generate the self-similar set.

Abstract. In [13], two of the authors gave a study of Lipschitz equivalence of selfsimilar sets through the augmented trees, a class of hyperbolic graphs introduced by Kaimanovich [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.

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