Subshifts with sparse traces

We study two-dimensional subshifts whose horizontal trace (a.k.a. projective subdynamics) contains only points of finite support. Our main result is a classification result for such subshifts satisfying a minimality property. As corollaries, we obtain new proofs for various known results on traces of SFTs, nilpotency and decidability of cellular automata, topological full groups and the subshift of prime numbers. We also construct various (sofic) examples illustrating the concepts.