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

 

Unstable attractors in manifolds


Author: J. J. Sánchez-Gabites
Journal: Trans. Amer. Math. Soc. 362 (2010), 3563-3589
MSC (2000): Primary 54H20, 57N65, 34D45
Published electronically: February 12, 2010
MathSciNet review: 2601600
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Abstract: Assume that $ K$ is a compact attractor with basin of attraction $ \mathcal{A}(K)$ for some continuous flow $ \varphi$ in a space $ M$. Stable attractors are very well known, but otherwise (without the stability assumption) the situation can be extremely wild. In this paper we consider the class of attractors with no external explosions, where a mild form of instability is allowed.

After obtaining a simple description of the trajectories in $ \mathcal{A}(K) - K$ we study how $ K$ sits in $ \mathcal{A}(K)$ by performing an analysis of the Poincaré polynomial of the pair $ (\mathcal{A}(K),K)$. In case $ M$ is a surface we obtain a nice geometric characterization of attractors with no external explosions, as well as a converse to the well known fact that the inclusion of a stable attractor in its basin of attraction is a shape equivalence. Finally, we explore the strong relations which exist between the shape (in the sense of Borsuk) of $ K$ and the shape (in the intuitive sense) of the whole phase space $ M$, much in the spirit of the Morse-Conley theory.


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Additional Information

J. J. Sánchez-Gabites
Affiliation: Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Address at time of publication: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: jaigabites@mat.ucm.es

DOI: http://dx.doi.org/10.1090/S0002-9947-10-05061-0
PII: S 0002-9947(10)05061-0
Received by editor(s): April 9, 2008
Published electronically: February 12, 2010
Additional Notes: This paper was written under partial support by Direccioń General de Investigacioń
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.