# Weakly-wandering set

*for an invertible measurable transformation $T$ of a measurable space $(X,\mathfrak B)$*

A measurable subset $A\subset X$ for which there is an infinite sequence of integers $n_i$ such that the sets $T^{n_i}$ are mutually disjoint (here, invertibility of $T$ is understood to mean existence and measurability of $T^{-1}$). If $T$ has a $\sigma$-finite quasi-invariant measure $\mu$ (defined on $\mathfrak B$), then a necessary and sufficient condition for $T$ to have a finite invariant measure equivalent to $\mu$ is that $\mu A=0$ for any weakly-wandering set $A$.

A word of warning. In topological dynamics there is a notion of weakly non-wandering point (see [3]) which has nothing to do with the notion defined above.

#### References

[1] | A.B. Hajian, S. Kakutani, "Weakly wandering sets and invariant measures" Trans. Amer. Math. Soc. , 110 : 1 (1964) pp. 136–151 |

[2] | A. Hajian, Y. Itô, "Weakly wandering sets and invariant measures for a group of transformations" J. Math. Mech. , 18 : 12 (1969) pp. 1203–1216 |

[3] | D.V. Anosov, I.V. Bronshtein, "Topological dynamics" , Dynamical Systems I , Encycl. Math. Sci. , I , Springer (1988) pp. 197–219 (Translated from Russian) |

**How to Cite This Entry:**

Weakly-wandering set.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Weakly-wandering_set&oldid=33108