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

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 [Luo and Lau, Adv. Math. 235 (2013) 555–579]. Our study extends the previous works.

On the Lipschitz equivalence of self‐affine sets

“…If a hyperbolic graph induced by an IFS is of bounded degree, then Theorem 5.5 of shows that (or the fractal K ) is totally disconnected if and only if the sizes of horizontal components in are uniformly bounded . Under the present setting, the statement can be strengthened as the following version.…”

“…The study of Lipschitz equivalence on Cantor sets was initiated by Copper and Pignataro and Falconer and Marsh . Along this line, it has been undergoing a great development . Up to now, thanks to many authors, the following one may be an elegant result on the Lipschitz equivalence of self‐similar sets.…”

“…In studying the Lipschitz equivalence of self‐similar sets, the author and his collaborators developed a technique of augmented trees . An augmented tree is defined on the symbolic space of the self‐similar IFS by adding more edges, and it is a Gromov hyperbolic graph .…”

“…Theorem would be false once removing the OSC in the assumption. As for the case without OSC, some deeper discussions will be necessary .…”

Algebraic criteria for Lipschitz equivalence of dust‐like self‐similar sets

“…Rao, Ruan and Xi [22] gave an affirmative answer to this question by introducing the graph‐directed technique to prove certain self‐similar sets being Lipschitz equivalence. Although considerable efforts [5, 10, 15–18, 20–25, 27–38] have been made in the study of this issue, it is still a long way to understand the Lipschitz equivalence of self‐similar sets.…”