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

 

An open set of maps for which every point
is absolutely nonshadowable


Authors: Guo-Cheng Yuan and James A. Yorke
Journal: Proc. Amer. Math. Soc. 128 (2000), 909-918
MSC (1991): Primary 58F13; Secondary 58F12, 58F14, 58F15
Published electronically: May 6, 1999
MathSciNet review: 1623005
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Abstract: We consider a class of nonhyperbolic systems, for which there are two fixed points in an attractor having a dense trajectory; the unstable manifold of one has dimension one and the other's is two dimensional. Under the condition that there exists a direction which is more expanding than other directions, we show that such attractors are nonshadowable. Using this theorem, we prove that there is an open set of diffeomorphisms (in the $C^{r}$-topology, $r > 1$) for which every point is absolutely nonshadowable, i.e., there exists $\epsilon > 0$ such that, for every $\delta > 0$, almost every $\delta$-pseudo trajectory starting from this point is $\epsilon$-nonshadowable.


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Additional Information

Guo-Cheng Yuan
Affiliation: Institute for Physical Science and Technology, and Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: gcyuan@ipst.umd.edu

James A. Yorke
Affiliation: Institute for Physical Science and Technology, and Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: yorke@ipst.umd.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05038-8
PII: S 0002-9939(99)05038-8
Received by editor(s): November 6, 1997
Received by editor(s) in revised form: April 21, 1998
Published electronically: May 6, 1999
Additional Notes: This research was supported by the National Science Foundation and Department of Energy.
Communicated by: Mary Rees
Article copyright: © Copyright 1999 American Mathematical Society