Substitution rules for aperiodic sequences of the cut and project type

We consider one-dimensional aperiodic sequences arising from a cut-and-project scheme with quadratic unitary Pisot numbers β. A construction of the substitution rule is described under rather general assumptions. It allows one to build a given cut-and-project sequence Σβ(Ω) starting from its arbitrary point. For a sequence with a convex acceptance window interval Ω, we prove that a substitution rule exists precisely if the boundary points of Ω are in the corresponding quadratic field [β]. Typically such a substitution has a reducible characteristic polynomial. Our main result is an algorithm for construction of such a substitution rule. Some examples are shown.