## 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. 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.