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

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.