Geodesic
A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources
Christian Bonatti1 ; Lorenzo Díaz2 ; Enrique R. Pujals3
1 Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon, France
2 Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro (PUC-RJ), Rio de Janeiro, Brazil
3 IMPA - Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
Annals of mathematics, Tome 158 (2003) no. 2, pp. 355-418.

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We show that, for every compact $n$-dimensional manifold, $n\geq 1$, 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 Mañé [Ma3]). In particular, we show that any $C^1$-robustly transitive diffeomorphism admits a dominated splitting.
DOI : 10.4007/annals.2003.158.355

Christian Bonatti 1 ; Lorenzo Díaz 2 ; Enrique R. Pujals 3

1 Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon, France
2 Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro (PUC-RJ), Rio de Janeiro, Brazil
3 IMPA - Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brazil 
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     title = {A $C^1$-generic dichotomy for diffeomorphisms: {Weak} forms of hyperbolicity or infinitely many sinks or sources},
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Christian Bonatti; Lorenzo Díaz; Enrique R. Pujals. A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources. Annals of mathematics, Tome 158 (2003) no. 2, pp. 355-418. doi : 10.4007/annals.2003.158.355. http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.158.355/

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