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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Hyperbolic limit sets


Author: Sheldon E. Newhouse
Journal: Trans. Amer. Math. Soc. 167 (1972), 125-150
MSC: Primary 58F15; Secondary 34C35
MathSciNet review: 0295388
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Abstract: Many known results for diffeomorphisms satisfying Axiom A are shown to be true with weaker assumptions. It is proved that if the negative limit set $ {L^ - }(f)$ of a diffeomorphism f is hyperbolic, then the periodic points of f are dense in $ {L^ - }(f)$. A spectral decomposition theorem and a filtration theorem for such diffeomorphisms are obtained and used to prove that if $ {L^ - }(f)$ is hyperbolic and has no cycles, then f satisfies Axiom A, and hence is $ \Omega $-stable. Examples are given where $ {L^ - }(f)$ is hyperbolic, there are cycles, and f fails to satisfy Axiom A.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0295388-6
PII: S 0002-9947(1972)0295388-6
Keywords: Limit set, hyperbolic, periodic point, topologically transitive, filtration, basic set, stable manifold, nonwandering
Article copyright: © Copyright 1972 American Mathematical Society