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

 

On the existence of attractors


Authors: Christian Bonatti, Ming Li and Dawei Yang
Journal: Trans. Amer. Math. Soc. 365 (2013), 1369-1391
MSC (2010): Primary 37B20, 37B25, 37C05, 37C10, 37C20, 37C29, 37C70, 37D05, 37D30, 37G25
Published electronically: August 22, 2012
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Abstract: On every compact $ 3$-manifold, we build a non-empty open set $ \mathcal U$ of $ \operatorname {Diff}^1(M)$ such that, for every $ r\geq 1$, every $ C^r$-generic diffeomorphism $ f\in \mathcal U\cap \operatorname {Diff}^r(M)$ has no topological attractors. On higher-dimensional manifolds, one may require that $ f$ has neither topological attractors nor topological repellers. Our examples have finitely many quasi-attractors. For flows, we may require that these quasi-attractors contain singular points. Finally we discuss alternative definitions of attractors which may be better adapted to generic dynamics.


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Additional Information

Christian Bonatti
Affiliation: Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon 21004, France
Email: bonatti@u-bourgogne.fr

Ming Li
Affiliation: School of Mathematical Sciences, Nankai University, Tianjin 300071, People’s Republic of China
Email: limingmath@nankai.edu.cn

Dawei Yang
Affiliation: School of Mathematics, Jilin University, Changchun 130000, People’s Republic of China
Email: yangdw1981@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05644-3
PII: S 0002-9947(2012)05644-3
Received by editor(s): March 20, 2010
Received by editor(s) in revised form: April 15, 2011
Published electronically: August 22, 2012
Additional Notes: This work was done during the stays of the second and third authors at the IMB, Université de Bourgogne, and they thank the IMB for its warm hospitality. The second author was supported by a postdoctoral grant of the Région Bourgogne, and the third author was supported by CSC of Chinese Education Ministry. This is a part of the third author’s Ph.D. thesis at Peking University.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.