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

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  • 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 $P$.
    Mathematics Subject Classification: Primary: 37C20, 37C29; Secondary: 37G25, 37D30.


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