Abstract: 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 $\mu$ 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 $\mu$ and their positive Lyapunov exponents approximate the positive Lyapunov exponents of $\mu$.

Tencent Meeting

ID: 358254855