Ergodic Theory and Dynamical Systems



Perturbations of the derivative along periodic orbits


CHRISTIAN BONATTI a1, NIKOLAS GOURMELON a1 and THÉRÈSE VIVIER a1
a1 IMB, UMR 5584 du CNRS, BP 47 870, 21078 Dijon Cedex, France (e-mail: bonatti@u-bourgogne.fr, nikolas.gourmelon@u-bourgogne.fr, therese.vivier@polytechnique.org)

Article author query
bonatti c   [Google Scholar] 
gourmelon n   [Google Scholar] 
vivier t   [Google Scholar] 
 

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.

(Published Online September 11 2006)
(Received October 1 2004)
(Revised January 8 2006)