# The Lyapunov exponents of generic volume-preserving and symplectic maps | Annals of Mathematics

Abstract We show that the integrated Lyapunov exponents of C1 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 C1 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.

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### 数学年刊（Annals of Mathematics）

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