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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
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