An Example of a Universally Observable Flow on the Torus

Abstract

In this paper we examine the question of existence of a two-dimensional universally observable system, i.e., dynamics which are observable by every continuous nonconstant real-valued function on the state space. We are motivated by the work of D. McMahon, 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. We then use these conditions to show the existence of a continuous universally observable flow on the torus. The proofs involve techniques and concepts from topological dynamics and dynamical systems on the torus.