On periodic solutions of 2-d linear difference equations

We study systems of 2-d linear difference equations with constant coefficients in a commutative quasi-Frobenius ring F, that is, F is Noetherian and self-injective. For instance, F could be a field or a residue class ring of the integers. Given a pair of positive integers p = (p₁; p₂), we first answer the following basic questions: Does there exist a p-periodic solution? When are all solutions p-periodic? Then we address the more interesting question of how to determine candidates for the period p. We characterize finitely generated systems, in which every trajectory is uniquely determined by finitely many initial values. If F is finite, all trajectories of a finitely generated system eventually become periodic, and we characterize the case where they are purely periodic (without pre-period), as well as the (component-wise) minimal period in this case.