Substitution rules for aperiodic sequences of the cut and project type

We consider one-dimensional aperiodic sequences arising from a cut-and-proj ect scheme with quadratic unitary Pisot numbers beta. A construction of the substitution rule is described under rather general assumptions. It allows one to build a given cut-and-project sequence Sigma (beta)(Omega) starting from its arbitrary point. For a sequence with a convex acceptance window i nterval Omega, we prove that a substitution rule exists precisely if the bo undary points of Omega are in the corresponding quadratic field Q[beta]. Ty pically such a substitution has a reducible characteristic polynomial. Our main result is an algorithm for construction of such a substitution rule. S ome examples are shown.