TY - JOUR

T1 - C1-generic symplectic diffeomorphisms

T2 - Partial hyperbolicity and zero centre Lyapunov exponents

AU - Bochi, Jairo

N1 - Funding Information:
Proof. Let k(f) denote the number of bundles in the finest dominated splitting of a map f : M → M. Then the Oseledets splitting at any regular point for f has at least k(f) bundles. Now let f ∈ PHω1(M) satisfy the generic properties from Proposition 7.4 and Theorem 1.1. That is, for almost every x ∈ M , the orbit of x is dense and the Oseledets splitting along it is (non-trivial and) dominated. The Oseledets splitting along the orbit of any such point extends to a dominated splitting over M, and hence must have exactly k(f) bundles. □ Proof of Theorem 1.3. If f belongs to the residual set given by Theorem 7.5 then the Oseledets space corresponding to zero exponents (if they exist) coincides almost everywhere with the ‘middle’ bundle of the finest dominated splitting, which by Theorem 2.2 is the centre bundle of a partially hyperbolic splitting. □ Acknowledgements. This work was partially supported by a CNPq–Brazil research grant.

PY - 2010/1

Y1 - 2010/1

N2 - We prove that if f is a C1-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if f is not Anosov then all the exponents in the centre bundle vanish. This establishes in full a result announced by Maé at the International Congress of Mathematicians in 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.

AB - We prove that if f is a C1-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if f is not Anosov then all the exponents in the centre bundle vanish. This establishes in full a result announced by Maé at the International Congress of Mathematicians in 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.

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U2 - 10.1017/S1474748009000061

DO - 10.1017/S1474748009000061

M3 - Article

AN - SCOPUS:77952551461

SN - 1474-7480

VL - 9

SP - 49

EP - 93

JO - Journal of the Institute of Mathematics of Jussieu

JF - Journal of the Institute of Mathematics of Jussieu

IS - 1

ER -