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

 

Diffeomorphisms satisfying the specification property


Authors: Kazuhiro Sakai, Naoya Sumi and Kenichiro Yamamoto
Journal: Proc. Amer. Math. Soc. 138 (2010), 315-321
MSC (2000): Primary 37A25, 37Bxx, 37C50, 37D20, 37D30
Published electronically: September 2, 2009
MathSciNet review: 2550197
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Abstract: Let $ f$ be a diffeomorphism of a closed $ C^\infty$ manifold $ M$. In this paper, we introduce the notion of the $ C^1$-stable specification property for a closed $ f$-invariant set $ \Lambda$ of $ M$, and we prove that $ f_{\vert\Lambda}$ satisfies a $ C^1$-stable specification property if and only if $ \Lambda$ is a hyperbolic elementary set. As a corollary, the $ C^1$-interior of the set of diffeomorphisms of $ M$ satisfying the specification property is characterized as the set of transitive Anosov diffeomorphisms.


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Additional Information

Kazuhiro Sakai
Affiliation: Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505, Japan
Email: kazsakai@cc.utsunomiya-u.ac.jp

Naoya Sumi
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
Email: sumi.n.aa@m.titech.ac.jp

Kenichiro Yamamoto
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
Email: yamamoto.k.ak@m.titech.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10085-0
PII: S 0002-9939(09)10085-0
Received by editor(s): February 6, 2009
Received by editor(s) in revised form: June 24, 2009
Published electronically: September 2, 2009
Additional Notes: The first author was supported by JSPS Grant-in-Aid for Scientific Research (C) (19540209).
Communicated by: Bryna Kra
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.