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 with continuous hyperbolic splitting $T_\Lambda M = E^s \oplus E^u$. For each sufficiently small $\varepsilon>0$ and each $x \in \Lambda$, let $W^s_\varepsilon(x)$ and $W^u_\varepsilon(x)$ denote the connected local stable and local unstable plaques of size $\varepsilon$ supplied by the [stable manifold theorem](/theorems/2778) for hyperbolic sets, chosen so that they vary continuously in the $C^1$ topology and, after shrinking the uniform plaque size if necessary, each whole plaque is represented in an adapted chart as a graph over the corresponding hyperbolic subspace.

There exists $\varepsilon_0 > 0$ such that for every $\varepsilon \in (0,\varepsilon_0]$ there exists $\delta = \delta(\varepsilon) > 0$ with the following property: if $x,y \in \Lambda$ and $d(x,y) < \delta$, then the full local plaques $W^u_\varepsilon(x)$ and $W^s_\varepsilon(y)$ intersect in exactly one point. This point is denoted

\begin{align*} [x,y] := W^u_\varepsilon(x) \cap W^s_\varepsilon(y). \end{align*}

Moreover, if $\Lambda$ is locally maximal, then $\varepsilon_0$ and $\delta(\varepsilon)$ may be chosen so that $[x,y] \in \Lambda$ for all such $x,y \in \Lambda$.