Geodesic
On isomorphity of measure-preserving $\mathbb Z^2$-actions that have isomorphic Cartesian powers
Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 8, pp. 193-212.

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Assume that $\Delta$ and $\Pi$ are representations of the group $\mathbb Z^2$ by operators on the space $L_2(X,\mu)$ that are induced by measure-preserving automorphisms, and for some $d$, the representations $\Delta^{\otimes d}$ and $\Pi^{\otimes d}$ are conjugate to each other, $\Delta\bigl(\mathbb Z^2\setminus(0,0)\bigr)$ consists of weakly mixing operators, and there is a weak limit (over some subsequence in $\mathbb Z^2$ of operators from $\Delta(\mathbb Z^2)$) which is equal to a nontrivial, convex linear combination of elements of $\Delta(\mathbb Z^2)$ and of the projection onto constant functions. We prove that in this case, $\Delta$ and $\Pi$ are also conjugate to each other. 
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A. E. Troitskaya. On isomorphity of measure-preserving $\mathbb Z^2$-actions that have isomorphic Cartesian powers. Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 8, pp. 193-212. http://geodesic.mathdoc.fr/item/FPM_2007_13_8_a11/

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