Let $(M,g)$ be a finite-dimensional smooth Riemannian manifold with associated distance function $d:M\times M\to[0,\infty)$, let $r\ge 1$, and let
paragraph
admin
\begin{align*}
f:M\to M
\end{align*}
latex_env
admin
be a $C^r$ diffeomorphism. Let $\Lambda\subset M$ be a compact $f$-invariant hyperbolic set, meaning that $f(\Lambda)=\Lambda$ and, for every $x\in\Lambda$, there is a direct-sum splitting of tangent spaces
such that $E^s=\bigsqcup_{x\in\Lambda}E^s_x\to\Lambda$ and $E^u=\bigsqcup_{x\in\Lambda}E^u_x\to\Lambda$ are continuous subbundles of $TM|_\Lambda$, the derivative maps satisfy $Df_x(E^s_x)=E^s_{f(x)}$ and $Df_x(E^u_x)=E^u_{f(x)}$ for every $x\in\Lambda$, and there are constants $A\ge 1$ and $0<\lambda<1$ such that for every $x\in\Lambda$, every integer $n\ge 0$, every $v_s\in E^s_x$, and every $v_u\in E^u_x$,
Here $Df_x^n:T_xM\to T_{f^n(x)}M$ denotes the derivative of the iterate $f^n$, and $Df_x^{-n}:T_xM\to T_{f^{-n}(x)}M$ denotes the derivative of the iterate $f^{-n}$.
paragraph
admin
Then there exist $\varepsilon>0$, constants $C'\ge 1$ and $0<\mu<1$, and, for every $x\in\Lambda$, embedded $C^r$ disks $W^s_\varepsilon(x)\subset M$ and $W^u_\varepsilon(x)\subset M$ such that $x\in W^s_\varepsilon(x)\cap W^u_\varepsilon(x)$,