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

TL;DR

This paper establishes a new framework using simple augmented trees to analyze the Lipschitz equivalence of self-similar sets, extending previous work by connecting hyperbolic graph theory with fractal geometry.

Contribution

It introduces simple augmented trees as Gromov hyperbolic graphs and proves their boundaries are Lipschitz equivalent to symbolic spaces of IFS, providing new criteria for self-similar sets.

Findings

Existence of near-isometry between augmented trees and symbolic spaces

Lipschitz equivalence of self-similar sets with or without open set condition

A criterion for self-similar sets to be Cantor-type sets

Abstract

Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graph on the IFS has been established recently. In the paper, we introduce a notion of simple augmented tree which is a Gromov hyperbolic graph. By generalizing a combinatorial device of rearrangeable matrix, we show that there exists a near-isometry between the simple augmented tree and the symbolic space of the IFS, so that their hyperbolic boundaries are Lipschitz equivalent. We then apply this to consider the Lipschitz equivalence of self-similar sets with or without assuming the open set condition. Moreover, we also provide a criterion for a self-similar set to be a Cantor-type set which completely answers an open question raised in \cite{LaLu13}. Our study extends the previous works.