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



Abundance of $ C^1$-robust homoclinic tangencies

Authors: Christian Bonatti and Lorenzo J. Díaz
Journal: Trans. Amer. Math. Soc. 364 (2012), 5111-5148
MSC (2010): Primary 37C05, 37C20, 37C25, 37C29, 37C70
Published electronically: May 24, 2012
MathSciNet review: 2931324
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Abstract: A diffeomorphism $ f$ has a $ C^1$-robust homoclinic tangency if there is a $ C^1$-neighborhood $ \mathcal {U}$ of $ f$ such that every diffeomorphism in $ g\in \mathcal {U}$ has a hyperbolic set $ \Lambda _g$, depending continuously on $ g$, such that the stable and unstable manifolds of $ \Lambda _g$ have some non-transverse intersection. For every manifold of dimension greater than or equal to three we exhibit a local mechanism (blender-horseshoes) generating diffeomorphisms with $ C^1$-robust homoclinic tangencies.

Using blender-horseshoes, we prove that homoclinic classes of $ C^1$-generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings (of appropriate dimensions) display $ C^1$-robust homoclinic tangencies.

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Additional Information

Christian Bonatti
Affiliation: Institut de Mathématiques de Bourgogne, B.P. 47 870, 21078 Dijon Cedex, France

Lorenzo J. Díaz
Affiliation: Departamento Matemática, PUC-Rio, Marquês de S. Vicente 225, 22453-900 Rio de Janeiro RJ, Brazil

Received by editor(s): September 29, 2009
Received by editor(s) in revised form: May 10, 2010, and August 4, 2010
Published electronically: May 24, 2012
Additional Notes: This paper was partially supported by CNPq, Faperj, and PRONEX (Brazil) and the Agreement in Mathematics Brazil-France. We acknowledge the warm hospitality of the I.M.P.A, the Institute de Mathématiques de Bourgogne, and PUC-Rio during the stays while preparing this paper
Dedicated: To Carlos Gutierrez (1944–2008), in memoriam
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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