Some new classes of directed graph IFSs

TL;DR

This paper introduces a new class of directed graph iterated function systems (IFSs) on the unit interval, removing previous restrictions and extending results to n-vertex systems with minimal conditions, broadening the understanding of their attractors.

Contribution

It presents a proof for 2-vertex directed graph IFSs without requiring rational independence or Hausdorff measure calculation, and extends these results to n-vertex systems with minimal assumptions.

Findings

Identified new classes of 2-vertex directed graph IFSs without previous restrictions.

Extended the results to n-vertex IFSs with only a gap length condition.

Provided a second result involving Hausdorff measure calculations for n-vertex systems.

Abstract

It has been shown that certain 2-vertex directed graph iterated function systems (IFSs), defined on the unit interval and satisfying the convex strong separation condition (CSSC), have attractors whose components are not standard IFS attractors where the standard IFSs may be with or without separation conditions. The proof required the multiplicative rational independence of parameters and the calculation of Hausdorff measure. In this paper we present a proof which does not have either of these requirements and so we identify a whole new class of 2-vertex directed graph IFSs. We extend this result to n-vertex (n>=2, CSSC) directed graph IFSs, defined on the unit interval, with no effective restriction on the form of the associated directed graph, subject only to a condition regarding level-1 gap lengths. We also obtain a second result for n-vertex (n>=2, CSSC) directed graph IFSs, defined on the unit interval, which does require the calculation of Hausdorff measure.