For a $C^\infty$ map on a compact manifold we prove that for a Lebesgue randomly picked point x there is an empirical measure from $x$ with entropy larger than or equal to the sum of positive Lyapunov exponents at $x$.

中文翻译：

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$$C^\infty $$C∞ 动力系统的物理量度熵

对于紧凑流形上的 $C^\infty$ 映射，我们证明对于 Lebesgue 随机选取的点 x，存在来自 $x$ 的经验测度，其熵大于或等于 $x$ 处的正李雅普诺夫指数之和。