  We study the decidability of the topological properties of some objects
coming from fractal geometry. We prove that having empty interior is
undecidable for the sets defined by two-dimensional graph-directed iterated
function systems. These results are obtained by studying a particular class of
self-affine sets associated with multi-tape automata. We first establish the
undecidability of some language-theoretical properties of such automata, which
then translate into undecidability results about their associated self-affine
sets.
