Flows on nonorientable 2-manifolds

Abstract

This paper extends the study initiated in [I] where a class ~4’ of orientable 2-manifolds, including all compact ones, was defined and it was shown that the central sequence (defined below) of any flow on any manifold in .4! terminates in at most two steps. It was also shown that any nonorientable 2-manifold supports a flow whose central sequence does not terminate in two steps. In this paper we define a class .4’of nonorientable 2-manifolds, including all compact ones, and show that the central sequence of any flow on a member of Nterminates in at most three steps. Preliminary results, mainly topological, are presented in Section 2 and in Section 3 the main results are established. The rest of this section contains the basic definitions and the notation of the paper. For more details and a brief history of the problem the reader is referred to [l] and [2]. Let (!zt 1 t E R> be a flow on a space X. By X1 we mean the set of all nonwandering points; thus, x E X1 if, and only if, for every neighborhood U of .T and every t E R, there exists s E R, s > t such that h,?(U) n U # p. Clearly Xi is closed in X and invariant under the flow. The central sequence (of the flow) is the possibly transfinite chain: Sr 2 X2 3 .** 2 X02 .*., where Xti = (X6)1 if 01 = /3 T 1 and if 01 is a limit ordinal, X0 = 0 {Xs 1 /3 C. CY’/. The center (of the flow) is nXa which may be described intrinsically in the following way. For each point f let Q(x) (A(x)) be the set of points y such that h,,(x) +y for some sequence t,A --z +m(t, --f -w). A point x is called Poisson stable in the positive (negative) direction provided x E 0(x)(x E A(x)) in which case we shall write, “x is P+(P-) stable.” A point which is both Pi and Pstable is called Poisson stable. It is known, Theorem 5.08 on p. 358 of [2], that the center is the closure of the set of Poisson stable points. The depth of the center is the least ordinal OL such that Xa = X=+l = “‘; that is, CY is the least ordinal such that Xb is the center. For the manifolds considered in this paper it will be shown the depth of the center of any flow is at most 3.