Global dominated splittings and the  Newhouse phenomenon |
| |
| Authors: | Flavio Abdenur Christian Bonatti Sylvain Crovisier |
| |
| Institution: | IMPA, Estrada D. Castorina 110, Jardim Botânico, 22460-010 Rio de Janeiro RJ, Brazil ; CNRS - Institut de Mathématiques de Bourgogne, UMR 5584, BP 47 870, 21078 Dijon Cedex, France ; CNRS - Laboratoire Analyse, Géométrie et Applications, UMR 7539, Université Paris 13, Avenue J.-B. Clément, 93430 Villetaneuse, France |
| |
| Abstract: | We prove that given a compact  -dimensional boundaryless manifold  ,  , there exists a residual subset  of the space of  diffeomorphisms  such that given any chain-transitive set  of  , then either  admits a dominated splitting or else  is contained in the closure of an infinite number of periodic sinks/sources. This result generalizes the generic dichotomy for homoclinic classes given by Bonatti, Diaz, and Pujals (2003). It follows from the above result that given a  -generic diffeomorphism  , then either the nonwandering set  may be decomposed into a finite number of pairwise disjoint compact sets each of which admits a dominated splitting, or else  exhibits infinitely many periodic sinks/sources (the `` Newhouse phenomenon"). This result answers a question of Bonatti, Diaz, and Pujals and generalizes the generic dichotomy for surface diffeomorphisms given by Mañé (1982). |
| |
| Keywords: | Dominated splitting Newhouse phenomenon $C^1$-generic dynamics |
|
| 点击此处可从《Proceedings of the American Mathematical Society》浏览原始摘要信息 |
| 点击此处可从《Proceedings of the American Mathematical Society》下载免费的PDF全文 |
|
