In 1967 Smale proved that for diffeomorphisms on closed smooth manifolds, Axiom
A
{\text {A}}
and no cycles are sufficient conditions for
Ω
\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
A
{\text {A}}
(resp.
A’
{\text {A’}}
) and no cycles are necessary as well as sufficient for
Ω
\Omega
-stability of diffeomorphisms (resp. vectorfields). Franks and Guckenheimer have worked on the diffeomorphism problem by strengthening the definition of
Ω
\Omega
-stable diffeomorphisms. In this paper it will be shown that an analogous strengthening of
Ω
\Omega
-stable vectorfields forces Smale’s conditions to be necessary. The major result of this paper is the following THEOREM. If
(
Λ
,
L
)
(\Lambda ,L)
is a compact laminated set,
N
N
is a normal bundle to the lamination, and
f
f
is an absolutely and differentiably
L
L
-stable diffeomorphism of a closed smooth manifold then
(
id -
f
#
¯
)
:
C
0
(
N
)
→
C
0
(
N
)
({\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
Ω
\Omega
-stable vectorfields.