Plane flows with closed orbits

Abstract

1. Introduction. One-parameter flows in topological and measure spaces, and especially in 2- and 3-dimensional Euclidean spaces, occur often in the consideration of various mathematical structures. Much of differential equation theory is related to this concept, not to mention such literal applications as aero- or hydro-dynamics. The flow represents the points of the space as idealized particles moving under a group of transformations indexed by the real line (time). The transformations could be, for example, measure-preserving and continuous (as in incompressible viscous flow), or continuously differentiable (as in the solutions to certain classes of differential equations), or merely measure-preserving (as in stochastic processes). Going further, it is interesting to know what can be said of such flows under particular sets of hypotheses. In this paper, we will be concerned with continuous flows in the Euclidean plane. For each flow, we consider the set of points which are fixed under every transformation. This set is an invariant of the flow; flows which are homeomorphically equivalent have homeomorphic invariant sets. In a previous paper [ 1], we show that every closed set in the plane can be represented as the fixed point set of some continuous flow. The solution, as it is constructed there, has many orbits (or trajectories) slowing down and approaching the fixed point set asymptotically in time. The points to which they converge, which we may call stagnation points after the aerodynamic usage, proliferate in this example in a way which is not observed in, say, fluid dynamics. In fact, the prevalence of stagnation points creates the feeling that the question answered is the wrong question, in a certain important sense. It is thus natural to inquire into the behavior of flows in which there are few or no stagnation points. This could be guaranteed by the stronger assumption that every orbit is closed, and this is the condition we impose on the flows in this paper. We shall see that not all closed sets can be invariant sets of such flows and, in fact, we show that a countable set of points cannot be an invariant set if it has a limit point. In this paper, we shall give a topological characterization of the sets which are invariant sets of