Wandering set

Remarks

If $f$ is a homeomorphism, the images $f^n(W)$, $n \in \Z$, of a wandering set are pairwise disjoint. The set of wandering points is open. In measurable dynamics, let $T$ be a measurable map of a measure space $(X, \mathcal{A}, \mu)$. A measurable set $W$ of positive measure is called wandering when the sets $T^{-n}(W)$, $n \ge 0$, are pairwise disjoint. Example: for $f(x) = x + 1$ on $\R$, the set $W = (0, 1)$ has $f^n(W) = (n, n+1)$, which is disjoint from $W$ for $n \ge 1$; so every point of $\R$ is wandering. Non-example: for $f(x) = x/2$ on $\R$, any open interval $U$ containing $0$ satisfies $0 \in f^n(U) \cap U$ for all $n$, so $U$ is not wandering and $0$ is not a wandering point.