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On essential coexistence of zero and nonzero Lyapunov exponents

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  • We show that there exists a $C^\infty$ volume preserving diffeomorphism $P$ of a compact smooth Riemannian manifold $\mathcal{M}$ of dimension 4, which is close to the identity map and has nonzero Lyapunov exponents on an open and dense subset $\mathcal{G}$ of not full measure and has zero Lyapunov exponent on the complement of $\mathcal{G}$. Moreover, $P|\mathcal{G}$ has countably many disjoint open ergodic components.
    Mathematics Subject Classification: 37D25, 37D30.


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