A \$C^1\$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources | Annals of Mathematics

Abstract We show that, for every compact n-dimensional manifold, n≥1, there is a residual subset of Diff1(M) of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities: Either it is contained in the closure of an infinite set of sinks or sources (Newhouse phenomenon), or it presents some weak form of hyperbolicity called dominated splitting (this is a generalization of a bidimensional result of Mañé [Ma3]). In particular, we show that any C1-robustly transitive diffeomorphism admits a dominated splitting.

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数学年刊（Annals of Mathematics）

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