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- W1965505406 abstract "In 1967 Smale proved that for diffeomorphisms on closed smooth manifolds, Axiom A and no cycles are sufficient conditions for I2stability 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 (resp. A') and no cycles are necessary as well as sufficient for Q2-stability of diffeomorphisms (resp. vectorfields). Franks and Guckenheimer have worked on the diffeomorphism problem by strengthening the definition of fl-stable diffeomorphisms. In this paper it will be shown that an analogous strengthening of Q2-stable vectorfields forces Smale's conditions to be necessary. The major result of this paper is the following THEOREM. If (A, 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 (id f): C0(N) ) 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 Q-stable vectorfields." @default.
- W1965505406 created "2016-06-24" @default.
- W1965505406 creator A5080195712 @default.
- W1965505406 date "1975-01-01" @default.
- W1965505406 modified "2023-10-16" @default.
- W1965505406 title "Converses to the $Omega $-stability and invariant lamination theorems" @default.
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- W1965505406 doi "https://doi.org/10.1090/s0002-9947-1975-0380885-8" @default.
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