Wandering set definition

Question

I've seen two apparently different definitions and am wondering which is correct. A set $W$ is wandering if $\{T^{-k}W; k\in \mathbb{N}_0\}$ (resp. $\{T^{k}W; k\in \mathbb{N}_0\}$) are pairwise disjoint. While I'm at it, what about using the collection $\{T^{k}W; k\in \mathbb{Z}\}$? Is there a sense in which any of these definitions are equivalent?

Answer 1

In fact there are various other similar notions of wandering set, and none of them got unanimous use in all areas. In other words, it is really not possible to tell you which one is "correct".

For example, sometimes one requires that there is an open neighborhood $U$ of the set such that its iterates $T^k(U)$ and $T^l(U)$ are disjoint for $|k-l|$ sufficiently large, or we require that the set of element of the action for which the images are not disjoint from the initial set is relatively compact.