On the topology of the set of completely unstable flows

Abstract:

We show that: (1) on any open manifold other than the line or plane, there exist nonsingular flows with $\Omega \ne \emptyset$ which can be perturbed, in the strong ${C^r}$ topology (any r), to flows with $\Omega \ne \emptyset$, and that (2) on certain open 3-manifolds there exist flows with $\Omega \ne \emptyset$ which cannot be approximated, in the strong ${{\mathcal {C}}^1}$ topology, by flows satisfying both $\Omega \ne \emptyset$ and no ${{\mathcal {C}}^1}$ $\Omega$-explosions. These examples give partial negative answers to the conjecture of Takens and White, that the completely unstable flows with the strong ${{\mathcal {C}}^r}$ topology equal the closure of their interior.