Pisot substitutions and their associated tiles par

Let σ be a unimodular Pisot substitution over a d letter alphabet and let X1, . . . , Xd be the associated Rauzy fractals. In the present paper we want to investigate the boundaries ∂Xi (1 ≤ i ≤ d) of these fractals. To this matter we define a certain graph, the so-called contact graph C of σ. If σ satisfies Manuscrit reçu le 17 novembre 2004. The author was supported by project S8310 of the Austrian Science Foundation. 488 Jörg M. Thuswaldner a combinatorial condition called the super coincidence condition the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries ∂X1, . . . , ∂Xd. From this graph directed system we derive an easy formula for the fractal dimension of ∂Xi in which eigenvalues of the adjacency matrix of C occur. An advantage of the contact graph is its relatively simple structure, which makes it possible to construct it for large classes of substitutions at once. In the present paper we construct the contact graph explicitly for a class of unimodular Pisot substitutions related to β-expansions with respect to cubic Pisot units. In particular, we deal with substitutions of the form σ(1) = 1 . . . 1 } {{ } b times 2, σ(2) = 1 . . . 1 } {{ } a times 3, σ(3) = 1 where b ≥ a ≥ 1. It is well known that these substitutions satisfy the above mentioned super coincidence condition. Thus we can give an explicit formula for the fractal dimension of the boundaries of the Rauzy fractals related to these substitutions.