Gromov hyperbolic graphs arising from iterations

Abstract

For a contractive iterated function system (IFS), it is known that there is a natural hyperbolic graph structure (augmented tree) on the symbolic space of the IFS that reflects the relationship among neighboring cells, and its hyperbolic boundary with the Gromov metric is Hölder equivalent to the attractor K[14,26,28]. This setup was taken up to study the probabilistic potential theory on K[21,20], and the bi-Lipschitz equivalence on K[29]. In this paper, we formulate a broad class of hyperbolic graphs, called expansive hyperbolic graphs, to capture the most essential properties from the augmented trees and the hyperbolic boundaries (e.g., the special geodesics, bounded degree property, metric doubling property, and Hölder equivalence). We also study a new setup of “weighted” IFS and investigate its connection with the self-similar energy form in the analysis of fractals.