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 locally maximal hyperbolic set for $f$. Then there exist an open neighbourhood $U \subset M$ of $\Lambda$ with compact closure and a $C^1$ neighbourhood $\mathcal{V}$ of $f$ in $\operatorname{Diff}^1(M)$ such that

\begin{align*} \Lambda = \bigcap_{n \in \mathbb{Z}} f^n(U). \end{align*}

For every $g \in \mathcal{V}$, define the maximal invariant set of $g$ in $U$ by

\begin{align*} \Lambda_g := \bigcap_{n \in \mathbb{Z}} g^n(U). \end{align*}

Then $\Lambda_g$ is a compact locally maximal hyperbolic set for $g$. Moreover, after replacing $\mathcal{V}$ by a smaller $C^1$ neighbourhood of $f$ if necessary, for every $g \in \mathcal{V}$ there exists a homeomorphism

\begin{align*} h_g: \Lambda \to \Lambda_g \end{align*}

such that

\begin{align*} h_g \circ f = g \circ h_g. \end{align*}

Consequently $f|_\Lambda$ and $g|_{\Lambda_g}$ are topologically conjugate.