Dimension approximation of attractors of graph directed IFSs by self-similar sets

Abstract

We show that for the attractor $(K_1,\dots ,K_q)$ of a graph directed iterated function system, for each $1\le j\le q$ and $\epsilon >0$ there exits a self-similar set $K\subseteq K_j$ that satisfies the strong separation condition and ${\dim }_H{K}_j-\epsilon <{\dim }_{H}K$. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of $K$. Using this property as a `black box' we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets.