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Heteroclinic connections and genericity of infinitely many sinks and sources. (Connexions hétéroclines et généricité d’une infinité de puits et de sources.) (French) Zbl 0944.37012
Summary: We say that two hyperbolic periodic points of a diffeomorphism \(f\) are persistently connected if there is a neighbourhood of \(f\) having a dense subset of diffeomorphisms for which there is a transitive set containing these two points. We prove that two points are generically in the same transitive set if and only if they are persistently connected with their homoclinic class being equal. As a consequence, we get the local genericity phenomenon (coexistence of infinitely many sinks or sources) for \(C^1\)-diffeomorphisms of three manifolds.

MSC:
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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