The construction of a universally observable flow on the torus

We give a description of the construction of a class of dynamical systems on a two-dimensional torus which are universally observable, i.e., systems which are observable by every continuous nonconstant real-valued function on the torus. We are motivated by the work of McMahon (1987) who proved that a class of three-dimensional manifolds with horocycle flow have this property. We examine this example and are able to give sufficient conditions for a flow to be universally observable and then construct a flow on the torus which satisfies these conditions. The proofs involve techniques and concepts from topological dynamics, dynamical systems on the torus and number theory.