Mathematics – Dynamical Systems
Scientific paper
2002-06-18
Annals of Mathematics, 161 (2005), No. 3, 1423--1485
Mathematics
Dynamical Systems
56 pages, 2 figures
Scientific paper
We show that the integrated Lyapunov exponents of $C^1$ volume preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding Oseledets splitting is trivial (all Lyapunov exponents equal to zero) or else dominated (uniform hyperbolicity in the projective bundle) almost everywhere. We deduce a sharp dichotomy for generic volume preserving diffeomorphisms on any compact manifold: almost every orbit either is projectively hyperbolic or has all Lyapunov exponents equal to zero. Similarly, for a residual subset of all $C^1$ symplectic diffeomorphisms on any compact manifold, either the diffeomorphism is Anosov or almost every point has zero as a Lyapunov exponent, with multiplicity at least 2. Finally, given any closed group $G \subset GL(d,\mathrm{R})$ that acts transitively on the projective space, for a residual subset of all continuous $G$-valued cocycles over any measure preserving homeomorphism of a compact space, the Oseledets splitting is either dominated or trivial.
Bochi Jairo
Viana Marcelo
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