On the topology and geometry of universally observable systems

As part of a general program for obtaining observability criterion for nonlinear systems by examining observability of restrictions of the dynamics to invariant sets of ??-limit points, we are led to the question of which minimal (in the sense of topological dynamics) will be observable by which observation functions. One remarkable case recently discovered by McMahon is that of universally observable systems; i.e. dynamics which are observable by every nonconstant continuous function. In this paper, we examine necessary conditions for the existence of a universally observable system defined on a state space X. We prove that such a system is necessarily minimal and that, if X is smooth, then X is compact, connected with vanishing Euler characteristic. As a consequence of this and the classification, initiated by Poincare and Denjoy, of vector field on the two-torus we show that low dimensional universally observable systems are unexpectedly rare. Indeed McMahon´s example in three dimensions may be the lowest dimensional occurrence of this truly nonlinear phenomenon.