# zbMATH — the first resource for mathematics

Power weak mixing does not imply multiple recurrence in infinite measure and other counterexamples. (English) Zbl 1013.37005
According to the Furstenberg-Szemerédi ergodic theorem: if $$(X,{\mathcal B},\mu,T)$$ is a finite measure-preserving invertible system then for any positive integer $$k$$ and for any measurable set $$E$$ with $$\mu(E)>0$$ there exists an integer $$n\geq 1$$ such that $$\mu(E\cap T^{n}E\cap T^{2n}E\cap \dots \cap T^{kn}E)>0$$. This property is called $$k$$-recurrence, and if it holds for all $$k>0$$ then it is called multiple recurrence. The paper demonstrates that in the case of infinite measure-preserving invertible systems a power weak mixing of the system does not imply the multiple recurrence. Some other counterexamples corresponding to infinite measure-preserving invertible systems are considered as well.

##### MSC:
 37A40 Nonsingular (and infinite-measure preserving) transformations 37A05 Dynamical aspects of measure-preserving transformations 28D05 Measure-preserving transformations 37A25 Ergodicity, mixing, rates of mixing
Full Text: