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January  2010, 26(1): 1-42. doi: 10.3934/dcds.2010.26.1

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

Nikolaz Gourmelon 1,

1. 

Université Bordeaux 1, 351, cours de la Libération, F 33405 TALENCE cedex, France

Received  January 2008 Revised  September 2009 Published  October 2009

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 $P$.
Keywords: homoclinic tangency, homoclinic class., bifurcation, Dominated splitting.
Mathematics Subject Classification: Primary: 37C20, 37C29; Secondary: 37G25, 37D3.
Citation: Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1
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