On the number of recurrent orbit closures

Abstract

Our purpose is to determine the maximum number of distinct recurrent orbit closures which can occur in a continuous flow on a compact nonorientable surface. Let X be a surface and let R be the real numbers. By a continuous flow on X we mean a continuous mapping of X X R to X such that for all x in X and for all s, t in R we have xO=x and x(s+t) = (xs)t where xt denotes the image of (x, t). The orbit of x is defined by 0(x) = { xt: tE R }. We say x is positively recurrent if given any neighborhood U of x and any positive number T there exists t> T such that xte U. Negative recurrence is defined analogously. The point x is recurrent if it is both negatively and positively recurrent. If x is positively or negatively recurrent, then Cl(O(x)) contains a recurrent point y such that Cl(O(x)) = Cl(O(y)). This follows from a simple category argument. Given a continuous flow on X we are interested in the number of distinct sets of the form Cl(O(x)) where x is recurrent and not periodic. Such a point will be called a strictly recurrent point. A local cross section at y, an element of X, is a subset S of X containing y which is homeomorphic to a nondegenerate closed interval and for which there exists an e> 0 such that the map (x, t)-*xt is a homeomorphism of SX [-e, e] onto the closure of an open neighborhood of y. If y is an interior point of X and if y is not a fixed point, then there exists a local cross section at y [5 ]. We will now state three theorems which are essential in the proof of our theorem. The first theorem shows us that we do not have to worry about descending chains of recurrent orbit closures. It was proved by Maler [3] for compact orientable surfaces, but one easily reduces the nonorientable case to the orientable one by lifting the flow to the orientable double covering. Received by the editors July 31, 1969 and, in revised form, December 8, 1969. AMS Subject Classifications. Primary 5482; Secondary 3465.