Let $M$ be a smooth manifold equipped with a Riemannian distance $d$, let $f:M\to M$ be a $C^1$ diffeomorphism, and let $\Lambda\subset M$ be a compact hyperbolic $f$-invariant set. Then there is a neighbourhood $N$ of $\Lambda$ with the following property. For every sufficiently small $\delta>0$, there exists $\varepsilon>0$ such that every bi-infinite $\varepsilon$-pseudo-orbit $(x_n)_{n\in\mathbb Z}$ contained in $\Lambda$, meaning

\begin{align*} d(f(x_n),x_{n+1})<\varepsilon \end{align*}

for every $n\in\mathbb Z$, is $\delta$-shadowed by a true orbit: there exists $y\in N$ such that

\begin{align*} d(f^n(y),x_n)<\delta \end{align*}

for every $n\in\mathbb Z$. If $\Lambda$ is locally maximal, then $N$ and $\delta$ may be chosen so that the shadowing point $y$ lies in $\Lambda$.