Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

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

G. C. BOOREa1 and K. J. FALCONERa1

a1 Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland. e-mail:,


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.

(Received August 18 2011)

(Revised August 13 2012)

(Online publication October 31 2012)