Let f be a $C^2$ diffeomorphism on a compact manifold. Ledrappier and Young introduced entropies along unstable foliations for an ergodic measure $\mu $ . We relate those entropies to covering numbers in order to give a new upper bound on the metric entropy of $\mu $ in terms of Lyapunov exponents and topological entropy or volume growth of sub-manifolds. We also discuss extensions to the $C^{1+\alpha },\,\alpha>0$ , case.

中文翻译：

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不稳定流形的熵和体积增长

令f是紧流形上的 $C^2$ 微分同胚。Ledrappier 和 Young 引入了沿不稳定叶状体的熵，用于遍历测量 $\mu$ 。我们将这些熵与覆盖数联系起来，以便根据 Lyapunov 指数和拓扑熵或子流形的体积增长给出度量熵 $\mu$ 的新上限 。我们还讨论了 $C^{1+\alpha },\,\alpha>0$ 案例的扩展。