Converses to the -stability and invariant lamination theorems

Author:
Allan Gottlieb

Journal:
Trans. Amer. Math. Soc. **202** (1975), 369-383

MSC:
Primary 58F10

DOI:
https://doi.org/10.1090/S0002-9947-1975-0380885-8

MathSciNet review:
0380885

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Abstract | References | Similar Articles | Additional Information

Abstract: In 1967 Smale proved that for diffeomorphisms on closed smooth manifolds, Axiom and no cycles are sufficient conditions for -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 (resp. ) and no cycles are necessary as well as sufficient for -stability of diffeomorphisms (resp. vectorfields). Franks and Guckenheimer have worked on the diffeomorphism problem by strengthening the definition of -stable diffeomorphisms. In this paper it will be shown that an analogous strengthening of -stable vectorfields forces Smale's conditions to be necessary. The major result of this paper is the following THEOREM. *If is a compact laminated set, is a normal bundle to the lamination, and is an absolutely and differentiably -stable diffeomorphism of a closed smooth manifold then 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 -stable vectorfields.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0380885-8

Keywords:
-stability,
Axiom ,
invariant manifolds,
invariant lamination,
hyperbolic,
no cycles,
normal hyperbolicity,
lamination,
laminated set,
absolutely -stable,
strongly -stable,
differentiably -stable,
absolutely -stable,
strongly -stable,
differentiably -stable

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© Copyright 1975
American Mathematical Society