Lipschitz equivalence of self-similar sets and hyperbolic boundaries

TL;DR

This paper studies the Lipschitz equivalence of augmented trees associated with self-similar sets, providing insights into the geometric structure and classification of these fractals under Lipschitz mappings.

Contribution

It analyzes a class of augmented trees and their Lipschitz equivalence, advancing understanding of the geometric classification of self-similar sets.

Findings

Characterization of Lipschitz equivalence for certain augmented trees

Application to classification of totally disconnected self-similar sets

Connection between hyperbolic boundaries and fractal geometry

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.