  In this note we give a simple sufficient condition for an affine iterated
function system to admit an invariant affine subspace persistently with respect
to changes in the translation parameters. This yields further examples of
tuples of contracting linear maps which do not satisfy the conclusions of
Falconer's theorem on the Hausdorff dimension of almost every self-affine set.
We also obtain new examples of iterated function systems of similarity
transformations which cannot satisfy the open set condition for any choice of
translation parameters, and resolve a related question of Peres and Solomyak.
