We study a class of graph-directed iterated function systems on R with algebraic parameters, which we call algebraic GIFS. 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...
