infinite intersection of orbit of bijective map Let $A$ be a countable infinite set, $T$ a bijective map of $A$, $U$ be proper subset of $A$(also infinite). Consider $\cap_{n=-\infty}^{\infty}T^{n}(U)$, where $T^{0}(U)=U$, can the above intersection be empty, while $\cap_{n=-M}^{N}T^{n}(U)$ is not empty for any finite $M, N$?
I think it is impossible to have the infinite intersection empty while finite intersection is not, but I can't prove it.
 A: It is possible! Let $A$ be the countable union of increasingly large finite sets, say $A=\{ (m,n)\in \Bbb Z^2\colon n\ge m\ge 0 \}$ which is the countable union of the $k$th "rows" $A_k = A \cap (\Bbb Z\times\{k\})$. Also let $U=\{ (m,n)\in \Bbb Z^2\colon n\ge m\ge 1 \} \subset A$ be the countable union of its $k$th rows $U_k = U \cap (\Bbb Z\times\{k\})$ (noting that $U_0=\emptyset$).
Define $T(m,n) = \bigl( (m+1\mod n+1),n \bigr)$. In other words, $T$ is the result of stitching together bijections of each individual $A_k$, and its action on $A_k$ is simply to cyclically permute its $k+1$ elements.
If we restrict $T$ to a single $A_k$, it's easy to check (for integers $Y\le Z$) that $\bigcap_{n=Y}^Z T^n(U_k)$ is nonempty precisely when $Z-Y<k$.

*

*In particular, $\bigcap_{n=-\infty}^\infty T^n(U_k)=\emptyset$ for each $k$, and therefore $\bigcap_{n=-\infty}^\infty T^n(U)=\emptyset$.

*However, for every fixed $-M$ and $N$, we have that $\bigcap_{n=-\infty}^\infty T^n(U_k)$ is nonempty for any $k>M+N$, and therefore $\bigcap_{n=-M}^N T^n(U)$ is nonempty.

