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Genericity of zero Lyapunov exponents. (English) Zbl 1023.37006
The author proves a result announced but probably not proved by R. Mañé: In the space of the \(C^1\) area-preserving diffeomorphisms on a compact surface the set of diffeomorphisms that are either Anosov or have zero Lyapunov exponents a.e. is residual. He also shows that for any fixed ergodic dynamical system on a compact metric space there is a residual set of continuous \(\text{SL}(2,\mathbb{R})\)-cocycles which either are uniformly hyperbolic or have zero exponents a.e.

MSC:
37C20 Generic properties, structural stability of dynamical systems
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
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