We prove a shadowing lemma for nonuniformly hyperbolic maps in Hilbert spaces. As applications, we firstly prove that the positive Lyapunov exponents of a hyperbolic ergodic measure µ can be approximated by positive Lyapunov exponents of atomic measures on hyperbolic periodic orbits; secondly, give an upper estimation of metric entropy by using the exponential growth rate of the number of such periodic points that their atomic measures approximate µ and their positive Lyapunov exponents approximate the positive Lyapunov exponents of µ.

中文翻译：

---

希尔伯特空间中非均匀双曲映射的阴影

我们证明了希尔伯特空间中非均匀双曲映射的阴影引理。作为应用，我们首先证明双曲遍历测度的正 Lyapunov 指数µ可以通过双曲周期轨道上原子测量的正李亚普诺夫指数来近似；其次，通过使用其原子测量近似的周期点数量的指数增长率来给出度量熵的上限估计µ并且它们的正 Lyapunov 指数近似于以下的正 Lyapunov 指数µ。