Attractors for iterated function schemes on [0,1]N[0,1]N are exceptional

Let X=[0,1]NX=[0,1]N with S={S1,…,SM}S={S1,…,SM} a finite set of contraction maps from X to itself. A nonempty compact subset E of X   is an attractor for the iterated function scheme SS if E=⋃i=1MSi(E)=S(E). We show that the typical closed set in [0,1]N[0,1]N is not an attractor of any iterated function scheme.