On a family of IFSs whose attractors are not connected

For an infinite-dimensional Banach space X, S and T bounded linear operators from X to X such that ‖S‖,‖T‖<1 and w∈X, let us consider the IFS Sw=(X,f1,f2), where f1,f2:X→X are given by f1(x)=S(x) and f2(x)=T(x)+w, for all x∈X. We prove that if the operator S is finite-dimensional, then the set {w∈X|the attractor ofSw is not connected} is open and dense in X.