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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Converses to the $ \Omega $-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: 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_\char93 }} ):{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.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0380885-8
Keywords: $ \Omega$-stability, Axiom $ {\text{A}}$, invariant manifolds, invariant lamination, hyperbolic, no cycles, normal hyperbolicity, lamination, laminated set, absolutely $ \Omega$-stable, strongly $ \Omega$-stable, differentiably $ \Omega$-stable, absolutely $ L$-stable, strongly $ L$-stable, differentiably $ L$-stable
Article copyright: © Copyright 1975 American Mathematical Society

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