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

 

Infinitesimally stable endomorphisms


Author: Hiroshi Ikeda
Journal: Trans. Amer. Math. Soc. 344 (1994), 823-833
MSC: Primary 58F10; Secondary 58F15
MathSciNet review: 1250821
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Abstract: It is well known that infinitesimal stability of diffeomorphisms is an open property. However, infinitesimal stability of endomorphisms is not an open property. So we consider the interior of the set of all infinitesimally stable endomorphisms. We prove that if f belongs to the interior of the set of all infinitesimally stable endomorphisms, then f is $ \Omega $-stable. This means a generalization of Smale's $ \Omega $-stability theorem for diffeomorphisms. Moreover, it is proved that for Anosov endomorphisms structural stability is equivalent to lying in the interior of the set of infinitesimally stable endomorphisms.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1250821-2
PII: S 0002-9947(1994)1250821-2
Keywords: Infinitesimal stability, Axiom A, prehyperbolic set, $ \Omega $-stability
Article copyright: © Copyright 1994 American Mathematical Society