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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
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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Keywords: <IMG WIDTH="20" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img11.gif" ALT="$\Omega$">-stability, structural stability, hyperbolic structure, nonwandering set
Article copyright: © Copyright 1971 American Mathematical Society