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Transactions of the American Mathematical Society

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

 

 

Necessary conditions for stability of diffeomorphisms


Author: John Franks
Journal: Trans. Amer. Math. Soc. 158 (1971), 301-308
MSC: Primary 57.20
DOI: https://doi.org/10.1090/S0002-9947-1971-0283812-3
MathSciNet review: 0283812
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Abstract: S. Smale has recently given sufficient conditions for a diffeomorphism to be $ \Omega $-stable and conjectured the converse of his theorem. The purpose of this paper is to give some limited results in the direction of that converse. We prove that an $ \Omega $-stable diffeomorphism $ f$ has only hyperbolic periodic points and moreover that if $ p$ is a periodic point of period $ k$ then the $ k$th roots of the eigenvalues of $ df_p^k$ are bounded away from the unit circle. Other results concern the necessity of transversal intersection of stable and unstable manifolds for an $ \Omega $-stable diffeomorphism.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0283812-3
Keywords: $ \Omega $-stability, structural stability, hyperbolic structure, nonwandering set
Article copyright: © Copyright 1971 American Mathematical Society