Measure-wise disjoint Rauzy fractals with the same incidence matrix

Abstract

The properties of the intersection of Rauzy fractals associated with substitutions having the same incidence matrix have been studied by several authors. Different techniques have been introduced and used for this purpose, one of them is the balanced pair algorithm. In the present paper we explore the actual limitations of this algorithm. We show that the balanced pair algorithm is defined and terminates only when the intersection of the Rauzy fractals have non-empty interior and when this condition fails it cannot be used.