Nonorientable recurrence of flows and interval exchange transformations

Abstract

In [l], Peixoto proved that on an orientable surface, the Morse-Smale vector fields (see [2, p. 1181) are dense in the space of C’ vector fields, r = 1, 2, . . . . Moreover, they are those which are structurally stable. However, the method he used to prove this result does not apply to a nonorientable surface. For special reasons the same fact holds for nonorientable surfaces of genus < 3 (see [3,4]). Whether the Morse-Smale vector fields are dense on nonorientable surfaces of genus 24 is still an open question. In fact, Gutierrez [S] showed that M2 has at least one C” vector field with nonorientable (nontrivial) recurrent trajectories: a trajectory y such that if p E y then y ( p} has two connected components, y + and y , and, if p E S, S being a segment transverse to the flow, there exist connected components ab c y + -S and cdcy--S such that abuS and cduS contain a one-sided simple closed curve. The existence of such recurrent trajectories are the main obstacle in tackling the problem of density of the Morse-Smale vector fields. A vector field on a surface of genus n, gives rise-through “cut and paste”-to a vector field on surfaces of greater genus, this is not necessarily the case if we consider a surface of smaller genus. The aim of this paper is to prove (Theorem 2) the existence of nondenumerable many C” vector fields on any nonorientable surface of genus n 2 4 which have nonorientable dense trajectories. These examples are such that the minimum genus of a surface where they can be defined is n. The richness of these examples shows that an extension of Peixoto’s theorem for nonorientable surface of genus 24 does not envolve just vector fields on a torus with two cross-caps (genus 4), but other surfaces as well. On the other hand, for the class of C”