Converses to the $\Omega$-stability and invariant lamination theorems

Abstract:

In 1967 Smale proved that for diffeomorphisms on closed smooth manifolds, Axiom ${\text {A}}$ and no cycles are sufficient conditions for $\Omega$-stability and asserted the analogous theorem for vectorfields. Pugh and Shub have supplied a proof of the latter. Since then a major problem in dynamical systems has been Smale’s conjecture that Axiom ${\text {A}}$ (resp. ${\text {A’}}$) and no cycles are necessary as well as sufficient for $\Omega$-stability of diffeomorphisms (resp. vectorfields). Franks and Guckenheimer have worked on the diffeomorphism problem by strengthening the definition of $\Omega$-stable diffeomorphisms. In this paper it will be shown that an analogous strengthening of $\Omega$-stable vectorfields forces Smale’s conditions to be necessary. The major result of this paper is the following THEOREM. If $(\Lambda ,L)$ is a compact laminated set, $N$ is a normal bundle to the lamination, and $f$ is an absolutely and differentiably $L$-stable diffeomorphism of a closed smooth manifold then $({\text {id - }}\overline {{f_\# }} ):{C^0}(N) \to {C^0}(N)$ is surjective. If the lamination is just a compact submanifold, the theorem is already new. When applied to flows, this theorem gives the above result on $\Omega$-stable vectorfields.