the union of local stable manifolds along local unstable manifolds

Question

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.

Answer 1

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