## Abstract

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 are 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 set S ⊂ GL(d) satisfying an accessibility condition, for a residual subset of all continuous S-valued cocycles over any measure-preserving homeomorphism of a compact space, the Oseledets splitting is either dominated or trivial. The condition on S is satisfied for most common matrix groups and also for matrices that arise from discrete Schrödinger operators.

Original language | English (US) |
---|---|

Pages (from-to) | 1423-1485 |

Number of pages | 63 |

Journal | Annals of Mathematics |

Volume | 161 |

Issue number | 3 |

DOIs | |

State | Published - May 2005 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty