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Nonorientable recurrence of flows and interval exchange transformations. (English) Zbl 0643.58003
In his famous paper M. Peixoto [Topology 1, 101-120 (1962; Zbl 0107.071)] showed that on an orientable closed 2-manifold the Morse-Smale vector fields are structurally stable and dense in the C r topology. Unfortunately, Peixoto’s methods fail when the manifold is nonorientable. Thus, the still open question of the density of Morse-Smale vector fields on nonorientable manifolds with large genus has been an area of active research. The main difficulties arise from the existence of vector fields with nonorientable recurrent trajectories which are not removable under perturbation by Peixoto’s closing lemma. The existence of \(C^{\infty}\) vector fields with such trajectories was discussed by C. Gutierrez [Trans. Am. Math. Soc. 241, 311-320 (1978; Zbl 0396.58017)].
In the present paper the main result is the existence of a nondenumerable set of vector fields, defined of any nonorientable surface with genus \(\geq 4\), which have nonorientable dense trajectories. The proof is carried out using interval exchange transformations.
Reviewer: C.Chicone

MSC:
37C75 Stability theory for smooth dynamical systems
37D15 Morse-Smale systems
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