First there is an interesting observation:

Let be a homeo and be a recurrent point . Then for each .

Proof. To derive a contradiction let’s pick to be the minimal time such that .

We first check that .

So for some .

Let .

Then and we prove inductively for each :

**(*)**—-.

Hence .

Since , we see for some . This contradicts our choice of .

Then comes the main topic:

Assume that for some , . Then there is a stable manifold tangent to at some iterate with injective radius . Without loss of generality we assume that for each .

First approach. Let be the plaque family given by the dominated splitting that is almost tangent to and a stronger assumption that for each . Pick such that for each and each . Let . Then contracts $W(x,r_0)$ uniformly: . Then and hence . Inductively we have for each and for each . So is contained in the stable set of .

Second approach. We find a subsequence of integers such that Without loss of generality we assume that for each and each . This may be related to the so called hyperbolic times. Then starting at each point we pull back an almost center-stable manifold through to through with uniform size. Then have a subsequence that converge to a disk . It is routine to check that is contained in the stable set and get contracted exponentially fast.