On some dynamical subsets of the rauzy fractal

Abstract

Let ($\text{N}\text{̄}{}^{\mathrm{n}}$,(+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 $\text{N}\text{̄}{}^{\mathrm{n}}$ and a map h_n: C → Cⁿ such that the dynamical system (C, h_n) is semiconjugate to ($\text{N}\text{̄}{}^{\mathrm{n-1}}\text{,(+1)}{}_{\mathrm{n-1}}$). 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,h_n) cannot be realized on the (n − 1)-torus.