A C1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources

• 成果类型: Article
• 署名作者: Bonatti, C; Díaz, LJ; Pujals, ER
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2003.158.355
• 发表日期: 2003
• 页码: 355-418
• 关键词: central direction; srb measures; systems; STABILITY; points; Axiom

摘要

We show that, for every compact n-dimensional manifold, n greater than or equal to 1, there is a residual subset of Diff(1)(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 Mane [Ma3]). In particular, we show that any C-1-robustly transitive diffeomorphism admits a dominated splitting.