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

Abstract:

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