Tilings from non-Pisot unimodular matrices

Using the unimodular Pisot substitution of the free monoid on d letters, the existence of graph-directed self-similar sets fXigi1⁄41; 2;...; d satisfying the set equation (0.0.1) with the positive measure on the A-invariant contracting plane P is well-known, where A is the incidence matrix of the substitution. Moreover, under some conditions, the set fXigi1⁄41; 2;...; d is the prototile of the quasi-periodic tiling of P (see Figure 1). In this paper, even in the case of non-Pisot matrix A, the generating method of graphdirected self-similar sets and quasi-periodic tilings is proposed under the ‘‘blocking condition’’.