Lipschitz equivalence of self-similar sets and hyperbolic boundaries

Abstract

In [9] Kaimanovich introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown in [13] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently.

Advances in Mathematics, accepted (2012). 29 pages, 10 figures

References in corpus (2)

• Ahlfors-David regular sets and bilipschitz maps
• Lipschitz Equivalence Class, Ideal Class and the Gauss Class Number Problem

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• Self-similar sets, simple augmented trees, and their Lipschitz equivalence