We obtain a dichotomy for \(C^{1}\)-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé.

中文翻译：

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具有正度量熵的微分同构

我们获得了\（C ^ {1} \）-通用的，保持体积的微分的二分法：几乎每个点的所有Lyapunov指数都消失了，或者体积是遍历的且不均匀的Anosov（即，不均匀的双曲且分裂为稳定和不稳定的空间为主）。这就完成了里卡多·马涅（RicardoMañé）首先提出的一个程序。