Abundance of $C^1$-robust homoclinic tangencies
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- by Christian Bonatti and Lorenzo J. Díaz PDF
- Trans. Amer. Math. Soc. 364 (2012), 5111-5148 Request permission
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
- Email: bonatti@u-bourgogne.fr
- Lorenzo J. Díaz
- Affiliation: Departamento Matemática, PUC-Rio, Marquês de S. Vicente 225, 22453-900 Rio de Janeiro RJ, Brazil
- Email: lodiaz@mat.puc-rio.br
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 5111-5148
- MSC (2010): Primary 37C05, 37C20, 37C25, 37C29, 37C70
- DOI: https://doi.org/10.1090/S0002-9947-2012-05445-6
- MathSciNet review: 2931324
Dedicated: To Carlos Gutierrez (1944–2008), in memoriam