On some dynamical subsets of the rauzy fractal

AbstractLet (N̄n,(+1)n) be the adic system associated to the substitution: 1 → 12,…,(n − 1) → 1n, n → 1. In Sirvent (1996) it was shown that there exist a subset Cn of N̄n and a map hn: C → Cn such that the dynamical system (C, hn) is semiconjugate to (N̄n−1,(+1)n−1). In this paper we compute the Hausdorff and Billingsley dimensions of the geometrical realizations of the set Cn on the (n− l)-dimensional torus. We also show that the dynamical system (Cn,hn) cannot be realized on the (n − 1)-torus