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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the existence of attractors
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by Christian Bonatti, Ming Li and Dawei Yang PDF
Trans. Amer. Math. Soc. 365 (2013), 1369-1391 Request permission

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
  • MR Author ID: 763903
  • Email: limingmath@nankai.edu.cn
  • Dawei Yang
  • Affiliation: School of Mathematics, Jilin University, Changchun 130000, People’s Republic of China
  • Email: yangdw1981@gmail.com
  • 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.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 1369-1391
  • MSC (2010): Primary 37B20, 37B25, 37C05, 37C10, 37C20, 37C29, 37C70, 37D05, 37D30, 37G25
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05644-3
  • MathSciNet review: 3003268