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Codimension one structurally stable chain classes

Authors: Xiao Wen and Lan Wen
Journal: Trans. Amer. Math. Soc. 368 (2016), 3849-3870
MSC (2010): Primary 37D20, 37D30, 37C20
Published electronically: May 14, 2015
MathSciNet review: 3453359
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Abstract: The well-known stability conjecture of Palis and Smale states that if a diffeomorphism is structurally stable, then the chain recurrent set is hyperbolic. It is natural to ask if this type of result is true for an individual chain class, that is, whether or not every structurally stable chain class is hyperbolic. Regarding the notion of structural stability, there is a subtle difference between the case of a whole system and the case of an individual chain class. The latter is more delicate and contains additional difficulties. In this paper we prove a result of this type for the latter, with an additional assumption of codimension 1. Precisely, let $ f$ be a diffeomorphism of a closed manifold $ M$ and let $ p$ be a hyperbolic periodic point of $ f$ of index 1 or $ \dim M-1$. We prove if the chain class of $ p$ is structurally stable, then it is hyperbolic. Since the chain class of $ p$ is not assumed in advance to be locally maximal, and since the counterpart of it for the perturbation $ g$ is defined not canonically but indirectly through the continuation $ p_g$ of $ p$, the proof is quite delicate.

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Additional Information

Xiao Wen
Affiliation: School of Mathematics and System Science, Beihang University, Beijing 100191, People’s Republic of China

Lan Wen
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China

Keywords: Structural stability, chain component, homoclinic class, hyperbolicity
Received by editor(s): December 7, 2012
Received by editor(s) in revised form: September 12, 2013, and March 24, 2014
Published electronically: May 14, 2015
Additional Notes: This work was partially supported by the Balzan Research Project of J. Palis. The first author was supported by NSFC 11301018. The second author was supported by NSFC 11231001.
Article copyright: © Copyright 2015 American Mathematical Society

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