I have that $(X,d)$ is a compact metric space, $T$ is a homeomorphism of $X$ and $S \subset\mathbb{N}$. We also have that $\forall \epsilon > 0$ , $\exists x_{\epsilon} \in X $ and $ n \in S$ such that $d(T^{n}x_{\epsilon}, x_{\epsilon}) < \epsilon $.
Does this imply that $\forall $ open set $U \subset X$ , $\exists n \in S$ such that such that $T^{-n}U \cap U \neq\emptyset$?
I thought that this would be true and would not be very hard to show. Boi! I was wrong. I would appreciate any help.
When I couldn't succeed for any $S$ I thought it would be easier to show for $S = \mathbb{N}$ but I was wrong again.
Here is a counter-example:
Take $X$ to be the closed interval $[-1,1]$. We are going to divide the open interval $(-1,1)$ into countably many closed intervals, each identified with an integer, as follows:
First, for each $n \in \mathbb{N}$, let $p_n = \sum_{k=1}^n (\frac{1}{2})^k$. Then $\displaystyle \lim_{n\to \infty} p_n = 1$. Let $p_{-n} = -p_n$. Then this gives a countable collection of points identified with $\mathbb{Z}\setminus \{0\}$ which goes to $1$ as $n$ goes to infinity, and goes to $-1$ as $n$ goes to negative infinity.
We then divide $(-1,1)$ into the closed intervals given by neighboring points: $I_0 = [p_{-1},p_1] = [-\frac{1}{2},\frac{1}{2}]$, and for the rest we define $I_n$ for $n>0$ by $I_n = [p_n, p_{n+1}]$, and follow the symmetric procedure for the negative integers.
Now, we define $T$ as a function which homeomorphically maps $I_z$ onto $I_{z+1}$, and which fixes the endpoints $-1$ and $1$. This is indeed a homeomorphism: we can find intervals sufficiently close to $-1$ and $1$ which will stay within $\varepsilon$ of the endpoints for any $\varepsilon$, (and it is clear that $T^{-1}$ is just the mirror image of $T$). Also, the endpoints trivially work as $x_{\varepsilon}$ for any $\varepsilon$.
However, no $I_n$ ever returns to itself under a forward or backward orbit under $T$, so their interiors are open sets which are disjoint from all of their forward and backward images.
NB: In dynamical systems, sets with behavior like these intervals are called wandering sets, and they are very important and show up in many interesting systems.