# Generation of homoclinic tangencies by $C^1$-perturbations.

Abstract : Given a [C^1] -diffeomorphism [f] of a compact manifold, we show that if the stable/unstable dominated splitting along a saddle is weak enough, then there is a small [C^1] -perturbation that preserves the orbit of the saddle and that generates a homoclinic tangency related to it. Moreover, we show that the perturbation can be performed preserving a homoclinic relation to another saddle. We derive some consequences on homoclinic classes. In particular, if the homoclinic class of a saddle [P] has no dominated splitting of same index as [P] , then a [C^1] -perturbation generates a homoclinic tangency related to [P] .
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Journal articles

https://hal.archives-ouvertes.fr/hal-00489042
Contributor : Nicolas Gourmelon <>
Submitted on : Thursday, June 3, 2010 - 4:42:59 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM

### Citation

Nicolas Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations.. Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2010, 26 (1), pp.1-42. ⟨10.3934/dcds.2010.26.1⟩. ⟨hal-00489042⟩

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