A generalized shadowing lemma

In this paper, we prove a generalized shadowing lemma. Let $f \in$ Diff$(M)$. Assume that $\Lambda$ is a closed invariant set of $f$ and there is a continuous invariant splitting $T\Lambda M = E\oplus F$ on $\Lambda$. For any $\lambda \in (0, 1)$ there exist $L > 0, d_0> 0$ such that for any $d \in (0, d_0]$ and any $\lambda$-quasi-hyperbolic d-pseudoorbit $\{x_i, n_i\}_{i=-\infty}^\infty$, there exists a point $x$ which Ld-shadows $\{x_i, n_i\}_{i=-\infty}^\infty$. Moreover, if $\{x_i, n_i\}_{i=-\infty}^\infty$ is periodic, i.e., there exists an $m > 0$ such that $x_{i+m}= x_i$ and $n_{i+m} = n_i$ for all $i$, then the point $x$ can be chosen to be periodic.