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 S w = ( X , f 1 , f 2 ) , where f 1 , f 2 : X → X are given by f 1 ( x ) = S ( x ) and f 2 ( 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 of S w is not connected} is open and dense in X.