Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure

TL;DR

This paper identifies a new class of attractors arising from directed graph IFSs that are not representable by standard IFSs, and computes their exact Hausdorff measure, expanding understanding of fractal attractors.

Contribution

It introduces a class of directed graph IFS attractors not obtainable by standard IFSs and calculates their exact Hausdorff measure, revealing new fractal structures.

Findings

Certain directed graph IFS attractors cannot be generated by standard IFSs.

Exact Hausdorff measure of these attractors is explicitly calculated.

New class of fractal attractors with known measure is identified.

Abstract

For directed graph iterated function systems (IFSs) defined on R, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known.