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

TL;DR

This paper demonstrates that attractors of graph directed IFSs can be approximated by self-similar sets with strong separation, enabling analysis of their geometric properties and dimensions.

Contribution

It introduces a method to approximate attractors of graph directed IFSs with self-similar sets satisfying strong separation, facilitating broader geometric and dimensional analysis.

Findings

Approximation of attractors by self-similar sets with strong separation.

Application to projections, intersections, and sum-product problems.

Enhanced understanding of fractal dimensions in complex systems.

Abstract

We show that for the attractor $(K_{1},\dots,K_{q})$ of a graph directed iterated function system, for each $1\leq j\leq q$ and $\varepsilon>0$ there exits a self-similar set $K\subseteq K_{j}$ that satisfies the strong separation condition and $\dim_{H}K_{j}-\varepsilon<\dim_{H}K$. 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.