Non-self-similar sets in [0,1]N of arbitrary dimension

We consider [0,1]N, the unit cube of RN, N⩾1. Let S={S1,…,SM} be a finite set of contraction maps from X to itself. A non-empty subset E of X is an attractor (or an invariant set) for the iterated function system (IFS) S if E=∪i=1MSi(E). We construct, for each s∈]0,N], a nowhere dense perfect set E contained in [0,1]N, with Hausdorff dimension s, which is not an attractor for any iterated function system composed of weak contractions from [0,1]N to itself.