General dynamical systems do not necessarily have closed-form solutions. An exception are linear systems, whose dynamics acting on the Cartesian plane can be solved exactly. However, linear maps acting on the 2-torus are strongly chaotic (under a fairly unrestrictive condition) and this leads to more exotic dynamics. In this talk, we provide some background into linear dynamics on the torus before introducing a family of piecewise linear maps acting on the 2-torus which was previously studied by Lagarias and Rains over the Cartesian plane. We study this map because it can be seen to be the simplest family of non-linear area-preserving maps acting on the 2-torus. We study the distribution of the periodic orbits for this map on rational lattices.