Let $f:M\rightarrow M$ be a $C^2$ hyperbolic diffeomorphism on compact connected riemannian manifold $M$. then there are local stable and unstable manifolds at each point denoted by $W^s\_\delta(x), W^u\_\delta(x)$ for sufficiently small $\delta>0$. its well known that $W^s(x)=\cup\_{n\geq0}f^{-n}(W^s\_\delta(f^n(x))) , W^u(x)=\cup\_{n\geq0}f^{n}(W^u\_\delta(f^{-n}(x)))$ are absolutely continuous foliations on $M$.


An argument which i have met several times is that $\cup\_{x\in W^u\_\delta(y)}W^s\_\delta(x)$ contains an open ball $B(y, r)$ for some $r>0$.I can see this in a geometric interpretation but i can't proof it. i was wondering if someone could help me please.



the union of local stable manifolds along local unstable manifolds
The local product structure says that for hyperbolic diffeos (it works also for flows), given two points $z$ and $z'$ in a small neighbourhood of $y$, then $W^s\_\delta(z)$ and $W^u\_\delta(z')$ intersect at exactly one point, often denoted by $[z,z']$.


Apply this property on $B(y,r)$ for $r$ small enough to any $z\in B(y,r)$ and $z'=y$. 
Then $W^u\_\delta(y)$ and $W^s\_\delta(z)$ intersect at exactly one point, that you call $x$. 
It proves the desired property. 


Barbara


