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Global dominated splittings and the $ C^1$ Newhouse phenomenon


Authors: Flavio Abdenur, Christian Bonatti and Sylvain Crovisier
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
Published electronically: March 14, 2006
MathSciNet review: 2213695
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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).


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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
Email: crovisie@math.univ-paris13.fr

DOI: https://doi.org/10.1090/S0002-9939-06-08445-0
Keywords: Dominated splitting, Newhouse phenomenon, $C^1$-generic dynamics
Received by editor(s): September 21, 2004
Published electronically: March 14, 2006
Communicated by: Michael Handel
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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