  Using a deterministic version of the self-similar (or hierarchical, or
fixed-point ) method for constructing 2-dimensional subshifts of finite type
(SFTs), we construct aperiodic 2D SFTs with a unique direction of
non-expansiveness and prove that the emptiness problem of SFTs is undecidable
even in this restricted case. As an additional application of our method, we
characterize the sets of directions that can be the set of non-expansive
directions of 2D SFTs.
