Global dominated splittings and the C1 Newhouse phenomenom

Abstract : We prove that given a compact n-dimensional boundaryless manifold M, n >=2, there exists a residual subset R of the space of C1 diffeomorphisms Diff such that given any chain-transitive set K of f in R then either K admits a dominated splitting or else K is contained in the closure of an infinite number of periodic sinks/sources. This result generalizes the generic dichotomy for homoclinic classes in [BDP]. It follows from the above result that given a C1-generic diffeomorphism f then either the nonwandering set Omega(f) may be decomposed into a finite number of pairwise disjoint compact sets each of which admits a dominated splitting, or else f exhibits infinitely many periodic sinks/sources (the ``C1 Newhouse phenomenon"). This result answers a question in [BDP] and generalizes the generic dichotomy for surface diffeomorphisms in [M].
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Flavio Abdenur, Christian Bonatti, Sylvain Crovisier. Global dominated splittings and the C1 Newhouse phenomenom. Proceedings of the American Mathematical Society, American Mathematical Society, 2006, 134, pp.2229-2237. ⟨hal-00538125⟩

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