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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Applications of Mañé's $ C^2$ connecting lemma


Author: Shuhei Hayashi
Journal: Proc. Amer. Math. Soc. 138 (2010), 1371-1385
MSC (2010): Primary 37C20, 37C29, 37C40, 37D05, 37D20
Published electronically: December 4, 2009
MathSciNet review: 2578529
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Abstract: We consider a few applications of Mañé's $ C^2$ Connecting Lemma. These are the $ C^2$ creation of homoclinic points associated to a basic set (i.e., isolated transitive hyperbolic set), a $ C^2$ locally generic criterion to know whether a given point belongs to the stable set of hyperbolic homoclinic classes, and that measurably hyperbolic diffeomorphisms (i.e., having the closure of supports of all invariant measures as a countable union of disjoint basic sets) are $ C^2$ generically uniformly hyperbolic diffeomorphisms.


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Additional Information

Shuhei Hayashi
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo, Japan
Email: shuhei@ms.u-tokyo.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10148-X
PII: S 0002-9939(09)10148-X
Keywords: $C^2$ generic properties, connecting lemma, invariant measures, basic sets, homoclinic points, homoclinic classes, uniform hyperbolicity, Axiom A with no cycles.
Received by editor(s): March 15, 2009
Received by editor(s) in revised form: July 31, 2009
Published electronically: December 4, 2009
Communicated by: Bryna Kra
Article copyright: © Copyright 2009 American Mathematical Society