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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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