In Brin Stuck, the following Corollary 5.6.6 is left as an exercise, that of which I'm not sure on how to prove. The stable-manifold theorem states that there exists $\epsilon >0$ small enough such that local stable manifolds $W^s_\epsilon(x) = \{ y \in M: d(f^n(x), f^n(y)) < \epsilon \quad \forall n \geq 0\}$ is a $C^1$ embedded manifold for every $x \in \Lambda$ in the hyperbolic set. Now we're interesting in the global stable manifold (defined below). Proposition 5.6.5 is just a set theoretic result, while the Corollary that follows seems to be a differential geometry question. Note that there is a typo here, authors should have written that $W^s(x)$ is an immersed $C^1$ submanifold (not embedded). I'm trying to find a prove for this claim. As a matter of fact, one can see that $W^s(x)$ is defined as a increasing union of embedded $C^1$-manifolds ! I posted a question here where I asked whether or not an increasing sequence of embedded submanifold is still an embedded submanifold. That is not the case, but at the time I thought that the stable manifold was to be a embedded submanifold (following the typo). Therefore, the question remains open as to whether or not such an increasing sequence can end up being an immersed submanifold !
Maybe that's not the way to do it...
Anyway, thanks for the help !
