Global dominated splittings and the $C^1$ Newhouse phenomenon
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- by Flavio Abdenur, Christian Bonatti and Sylvain Crovisier PDF
- Proc. Amer. Math. Soc. 134 (2006), 2229-2237 Request permission
Abstract:
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 $\mathrm {Diff}^1(M)$ 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 Bonatti, Diaz, and Pujals (2003). 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 and generalizes the generic dichotomy for surface diffeomorphisms given by Mañé (1982).References
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Additional Information
- Flavio Abdenur
- Affiliation: IMPA, Estrada D. Castorina 110, Jardim Botânico, 22460-010 Rio de Janeiro RJ, Brazil
- Email: flavio@impa.br
- Christian Bonatti
- Affiliation: CNRS - Institut de Mathématiques de Bourgogne, UMR 5584, BP 47 870, 21078 Dijon Cedex, France
- Email: bonatti@u-bourgogne.fr
- Sylvain Crovisier
- Affiliation: CNRS - Laboratoire Analyse, Géométrie et Applications, UMR 7539, Université Paris 13, Avenue J.-B. Clément, 93430 Villetaneuse, France
- MR Author ID: 691227
- Email: crovisie@math.univ-paris13.fr
- Received by editor(s): September 21, 2004
- Published electronically: March 14, 2006
- Communicated by: Michael Handel
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 2229-2237
- MSC (2000): Primary 37D25, 37D30
- DOI: https://doi.org/10.1090/S0002-9939-06-08445-0
- MathSciNet review: 2213695