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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 Birkhoff centers


Author: I. P. Malta
Journal: Trans. Amer. Math. Soc. 262 (1980), 181-193
MSC: Primary 58F15; Secondary 58F20
MathSciNet review: 583851
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Abstract: The purpose of this paper is to show that if f is a diffeomorphism of a compact manifold whose Birkhoff center, $ c(f)$, is hyperbolic and has no cycles, then f satisfies Axiom A and is $ \Omega $-stable. To obtain a filtration for $ c(f)$, the concept of an isolated set for a homeomorphism of a compact metric space is introduced. As a partial converse it is proved that if $ c(f)$ is hyperbolic and f is $ \Omega $-stable, then $ c(f)$ has the no cycle property. A characterization of $ \Omega $-stability when $ c(f)$ is finite is also given.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0583851-4
PII: S 0002-9947(1980)0583851-4
Article copyright: © Copyright 1980 American Mathematical Society