The theory of Gromov hyperbolic graphs and its application to Lipschitz equivalence of self-similar sets
Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graphs on the IFS has been established recently. In this talk, we introduce a notion of simple augmented tree on the IFS which is a Gromov hyperbolic graph. By using a combinatoric device called rearrangeable matrix and its generalization, we show that there exists a near-isometry between the simple augmented tree and the symbolic space of the IFS, so that their boundaries are Lipschitz equivalent. We then apply this result to consider the Lipschitz equivalence of self-similar sets with or without the open set condition, which is an interesting topic in fractal geometry and geometric measure theory.