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Global dominated splittings and the $$C^1$$ Newhouse phenomenon. (English) Zbl 1088.37013
Summary: We prove that given a compact $$n$$-dimensional boundaryless manifold $$M$$, $$n \geq 2$$, there exists a residual subset $$\mathcal{R}$$ of the space of $$C^1$$ diffeomorphisms $$\text{Diff}^1(M)$$ such that given any chain-transitive set $$K$$ of $$f \in \mathcal{R}$$, 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 given by C. Bonatti, L. J. Diaz and E. R. Pujals [Ann. Math. (2) 158, 355–418 (2003; Zbl 1049.37011)].
It follows from the above result that given a $$C^1$$-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 “$$C^1$$ Newhouse phenomenon”). This result answers a question of Bonatti, Diaz and Pujals [loc. cit.] and generalizes the generic dichotomy for surface diffeomorphisms given by R. Mañé [Ann. Math. (2) 116, 503–540 (1982; Zbl 0511.58029)].

##### MSC:
 37D30 Partially hyperbolic systems and dominated splittings 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
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