Let $f: M \to M$ be a $C^r$ diffeomorphism with $r \ge 1$, and let $\Lambda \subset M$ be a compact hyperbolic set. There exist $\varepsilon>0$, constants $C'\ge 1$ and $0<\mu<1$, and for each $x\in\Lambda$ embedded $C^r$ disks $W^s_\varepsilon(x)$ and $W^u_\varepsilon(x)$ such that \begin{align*} T_x W^s_\varepsilon(x) = E^s_x, \qquad T_x W^u_\varepsilon(x) = E^u_x. \end{align*} The disks satisfy $f(W^s_\varepsilon(x))\subset W^s_\varepsilon(f(x))$ and $f^{-1}(W^u_\varepsilon(x))\subset W^u_\varepsilon(f^{-1}(x))$. If $y\in W^s_\varepsilon(x)$, then \begin{align*} d(f^n(y),f^n(x)) \le C'\mu^n d(y,x) \end{align*} for all $n\ge 0$, and the analogous backward estimate holds on $W^u_\varepsilon(x)$.