Abstract  We introduce a new model of weighted automata: the desert automata. We show that their limitedness problem is PSPACE-complete by solving the underlying Burnside problem. As an application of this result, we give a positive solution to the so-called finite substitution problem which was open for more than 10 years: given recognizable languages K and L, decide whether there exists a finite substitution σ such that σ(K) = L.