When does the stable set contain some disk?

First there is an interesting observation:
Let f:X\to X be a homeo and x\in X be a recurrent point x\in\omega(x,f). Then x\in\omega(x,f^n) for each n\ge1.
Proof. To derive a contradiction let’s pick n\ge2 to be the minimal time such that x\notin\omega(x,f^n).

We first check that \omega(x,f)=\bigcup_{0\le k\le n-1}\omega(f^kx,f^n).

So x\in\omega(f^kx,f^n) for some k\in\{1,\cdots,n-1\}.
Let l=n-k\in\{1,\cdots,n-1\}.

Then f^lx\in\omega(x,f^n) and we prove inductively f^{lj}x\in\omega(x,f^n) for each j\ge1:

(*)—-f^{l(j+1)}x=f^l(f^{lj}x)\in f^l\omega(x,f^n)=\omega(f^lx,f^n)\subset \omega(x,f^n).

Hence \omega(x,f^l)\subset\omega(x,f^n).

Since x\notin\omega(x,f^n), we see x\notin\omega(x,f^l) for some l<n. This contradicts our choice of n.

Then comes the main topic:

Assume that for some x\in M, \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\log\|Df|_{E(f^kx)}\|<-1. Then there is a stable manifold tangent to E at some iterate f^nx with injective radius r_0. Without loss of generality we assume that \frac{1}{n}\sum_{k=0}^{n-1}\log\|Df|_{E(f^kx)}\|<-1 for each n\ge1.

First approach. Let \mathcal{W} be the plaque family given by the dominated splitting that is almost tangent to E and a stronger assumption that \frac{1}{n}\sum_{k=0}^{n-1}\log\|Df|_{E(f^kx)}\|<-1 for each n\ge1. Pick r_0 such that \frac{\|Df|_{T_yW}\|}{\|Df|_{E(f^nx)\|}}\in(e^{-\delta},e^{\delta}) for each y\in W(f^nx,r_0) and each n\ge 0. Let \lambda=e^{\delta-1}<1. Then f contracts $W(x,r_0)$ uniformly: fW(x,r_0)\subset W(fx,\lambda r_0). Then \frac{\|Df|_{T_yW}\|\|Df|_{T_{fy}W}\|}{\|Df|_{E(x)}\|\|Df|_{E(fx)}\|}\le\lambda^2 and hence f^2W(x,r_0)\subset W(f^2x,\lambda^2 r_0). Inductively we have \frac{\|Df|_{T_yW}\|\cdots\|Df|_{T_{f^{n-1}y}W}\|}{\|Df|_{E(x)}\|\cdots\|Df|_{E(f^{n-1}x)}\|} \le\lambda^n for each and f^nW(x,r_0)\subset W(^nx,\lambda^n r_0) for each n\ge1. So W(x,r_0) is contained in the stable set of x.

Second approach. We find a subsequence of integers n_i such that Without loss of generality we assume that \sum_{k=j}^{n_i-1}\log\|Df|_{E(f^kx)}\|<-(n_i-j) for each j=0,\cdots, n_i-1 and each i\ge1. This may be related to the so called hyperbolic times. Then starting at each point f^{n_i}x we pull back an almost center-stable manifold D_{n_i} through f^{n_i}x to D_{n_i,0} through x with uniform size. Then D_{n_i,0} have a subsequence that converge to a disk D. It is routine to check that D is contained in the stable set and get contracted exponentially fast.

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