Heteroclinic connections and genericity of infinitely many sinks or sources

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 of the Newhouse's phenomenon (coexistence of infinitely many sinks or sources) for C-1-diffeomorphisms of three manifolds. (C) Elsevier, Paris.