Dual systems of algebraic iterated function systems

We study a class of graph-directed iterated function systems on R with algebraic parameters, which we call algebraic GIPS. We construct a dual IFS of an algebraic GIFS, and study the relations between the two systems. We determine when a dual system satisfies the open set condition, which is fundamental. For feasible Pisot systems, we construct the left and right Rauzy-Thurston tilings, and study their multiplicities and decompositions. We also investigate their relation with codings space, domain-exchange transformation, and the Pisot spectrum conjecture. The dual IFS provides a unified and simple framework for Rauzy fractals, beta-tilings and related studies, and allows us gain better understanding. (C) 2013 Elsevier Inc. All rights reserved.