zbMATH — the first resource for mathematics

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

37C75 Stability theory for smooth dynamical systems
37D15 Morse-Smale systems
Full Text: DOI
[1] Peixoto, M, Structure stability on two-dimensional manifolds, Topology, 1, 101-120, (1962) · Zbl 0107.07103
[2] Palis, J; de Melo, W, Geometric theory of dynamical systems, (1982), Springer-Verlag New York
[3] Markley, N, The PoincarĂ©-bendixon theorem for the Klein bottle, Trans. amer. math. soc., 135, 159-165, (1969) · Zbl 0175.50101
[4] Gutierrez, C, Structural stability on the torus with a cross-cap, Trans. amer. math. soc., 241, 311-320, (1978) · Zbl 0396.58017
[5] Gutierrez, C, Smooth nonorientable nontrivial recurrence on two-manifolds, J. differential equations, 29, 388-395, (1978) · Zbl 0413.58018
[6] Keane, M, Interval exchange transformations, Math. Z., 141, 25-31, (1975) · Zbl 0278.28010
[7] Cornfeld, I.P; Fomin, S.V; Sinai, Ya.G, Ergodic theory, (1982), Springer-Verlag New York · Zbl 0493.28007
[8] Masur, H, Interval exchange transformations and measured foliations, Ann. math., 115, 169-200, (1982) · Zbl 0497.28012
[9] Veech, W.A, Gauss measures for transformations on the space of interval exchange maps, Ann. math., 115, 201-242, (1982) · Zbl 0486.28014
[10] \scA. Nogueira, Almost all interval exchange transformations with flips are nonergodic, Ergodic Theory Dynamical Systems, to appear. · Zbl 0697.58027
[11] Gutierrez, C, Smoothability of Cherry flows on two-manifolds, (), 308-331
[12] Lancaster, P; Timenetsky, M, The theory of matrices, (1985), Academic Press New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.