Abstract

Let ((N)over bar (n)>(+1)(n)) be the adic system associated to the substitution: 1 --> 12,...,(n - 1) --> ln, n --> 1. In Sirvent (1996) it was shown that there exist a subset C-n of (N)over bar (n)> and a map h(n):C-n --> C-n such that the dynamical system (C-n, h(n)) is semiconjugate to ((N)over bar (n - 1)>, (+ 1)(n - 1)). In this paper we compute the Hausdorff and Billingsley dimensions of the geometrical realizations of the set C-n on the (n - 1)-dimensional torus. We also show that the dynamical system (C-n, h(n)) cannot be realized on the (n - 1)-torus.