Graph-Theoretic Characterization of Separation Conditions in Self-Affine Iterated Function Systems

For a self-affine iterated function system (IFS) {fj}j=1N on Rd defined by fj(x)=A−1(x+dj), where A is an expansive matrix and dj∈Rd, we reveal a novel characterization of the open set and weak separation conditions through the bounded degree of the augmented tree induced by the IFS. Furthermore, the augmented tree is shown to be a Gromov hyperbolic graph, and its hyperbolic boundary is Hölder equivalent to the self-affine set generated by the IFS, establishing a canonical association between self-affine IFSs and Gromov hyperbolic graphs.