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Perturbations of the derivative along periodic orbits

Published online by Cambridge University Press:  11 September 2006

CHRISTIAN BONATTI ,
NIKOLAS GOURMELON  and
THÉRÈSE VIVIER
CHRISTIAN BONATTI
Affiliation:
IMB, UMR 5584 du CNRS, BP 47 870, 21078 Dijon Cedex, France (e-mail: [email protected], [email protected], [email protected])
NIKOLAS GOURMELON
Affiliation:
IMB, UMR 5584 du CNRS, BP 47 870, 21078 Dijon Cedex, France (e-mail: [email protected], [email protected], [email protected])
THÉRÈSE VIVIER
Affiliation:
IMB, UMR 5584 du CNRS, BP 47 870, 21078 Dijon Cedex, France (e-mail: [email protected], [email protected], [email protected])
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Abstract

We show that a periodic orbit of large period of a diffeomorphism or flow either admits a dominated splitting of a prescribed strength or can be turned into a sink or a source by a $C^1$-small perturbation along the orbit. As a consequence we show that the linear Poincaré flow of a $C^1$-vector field admits a dominated splitting over any robustly transitive set.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 26 , Issue 5 , October 2006 , pp. 1307 - 1337
Copyright
2006 Cambridge University Press

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