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Heteroclinic cycles and homoclinic closures for generic diffeomorphisms. (English) Zbl 1034.37013
Summary: We prove some \(C^1\) generic results about orbit-connecting, in particular about heteroclinic cycles and homoclinic closures. As a consequence we obtain a three-ways \(C^1\) density theorem: Diffeomorphisms with either infinitely many weakly transitive components or a heterodimensional cycle are \(C^1\) dense in the complement of the \(C^1\) closure of Axiom A and no-cycle diffeomorphisms.

MSC:
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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