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

DOI:
https://doi.org/10.1090/S0002-9947-2012-05644-3

Published electronically:
August 22, 2012

MathSciNet review:
3003268

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Abstract: On every compact -manifold, we build a non-empty open set of such that, for every , every -generic diffeomorphism has no topological attractors. On higher-dimensional manifolds, one may require that 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:
https://doi.org/10.1090/S0002-9947-2012-05644-3

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.