Structural stability for flows on the torus with a cross-cap

Abstract

Let X r ( M ~ ) , r ≥ 1 {{\mathcal {X}}^r}({\tilde M}), r \geq 1 , denote the space of C r {C^r} -vector fields on the torus with a cross-cap M ~ \tilde M . We show that the Morse-Smale vector fields of X r ( M ~ ) {{\mathcal {X}}^r}({\tilde M}) are dense on it. We also give a simple proof that a C 0 {C^0} -flow on the Klein bottle cannot support a nontrivial ω \omega -recurrent trajectory.