Let $r \in \mathbb{N} \cup \{\infty\}$, let $M$ be a finite-dimensional second-countable Hausdorff $C^r$ manifold, and let $f: M \to M$ be a $C^r$ diffeomorphism. Let $p \in M$ satisfy $f(p)=p$. Suppose that $p$ is hyperbolic, meaning that the complexification of the [linear map](/page/Linear%20Map) $df_p: T_pM \to T_pM$ has no eigenvalue of modulus $1$. Let $T_pM = E^s \oplus E^u$ be the real $df_p$-invariant splitting into the generalized eigenspaces corresponding respectively to eigenvalues of modulus less than $1$ and eigenvalues of modulus greater than $1$.

Then there exist an open neighbourhood $U \subset M$ of $p$, an embedded $C^r$ disk $W^s_{\mathrm{loc}}(p) \subset U$ of dimension $\dim E^s$, and an embedded $C^r$ disk $W^u_{\mathrm{loc}}(p) \subset U$ of dimension $\dim E^u$ such that $p \in W^s_{\mathrm{loc}}(p) \cap W^u_{\mathrm{loc}}(p)$, $T_pW^s_{\mathrm{loc}}(p)=E^s$, and $T_pW^u_{\mathrm{loc}}(p)=E^u$.

After possibly replacing $U$, $W^s_{\mathrm{loc}}(p)$, and $W^u_{\mathrm{loc}}(p)$ by smaller neighbourhoods of $p$, one has $f(W^s_{\mathrm{loc}}(p)) \subset W^s_{\mathrm{loc}}(p)$ and $f^{-1}(W^u_{\mathrm{loc}}(p)) \subset W^u_{\mathrm{loc}}(p)$. Moreover, with $\mathbb{N}_0 := \{0,1,2,\dots\}$,

\begin{align*} W^s_{\mathrm{loc}}(p)=\{x \in U : f^n(x) \in U \text{ for every } n \in \mathbb{N}_0\} \end{align*}

and

\begin{align*} W^u_{\mathrm{loc}}(p)=\{x \in U : f^{-n}(x) \in U \text{ for every } n \in \mathbb{N}_0\}. \end{align*}

Finally, for every Riemannian distance $d$ defined on a neighbourhood of $p$, after shrinking $U$ if necessary, there exist constants $C>0$ and $\theta \in (0,1)$ such that, for every $x \in W^s_{\mathrm{loc}}(p)$ and every $n \in \mathbb{N}_0$,

\begin{align*} d(f^n(x),p) \le C\theta^n d(x,p), \end{align*}

and, for every $x \in W^u_{\mathrm{loc}}(p)$ and every $n \in \mathbb{N}_0$,

\begin{align*} d(f^{-n}(x),p) \le C\theta^n d(x,p). \end{align*}