Dual systems of algebraic iterated function systems

• Published in: Advances in mathematics (New York. 1965) Vol. 253; pp. 63 - 85
• Main Authors: Rao, Hui, Wen, Zhi-Ying, Yang, Ya-Min
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.03.2014
• Subjects: Iterated function system; Pisot spectrum conjecture; Rauzy fractal; Self-similar tiling; Iterated function system; Self-similar tiling; Rauzy fractal; Pisot spectrum conjecture
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2013.11.010

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 conjecture. The dual IFS provides a unified and simple framework for Rauzy fractals, β-tilings and related studies, and allows us gain better understanding.