On the isomorphism of tensor powers of ergodic flows

Abstract

The following question due to Thouvenot is well-known in ergodic theory. Let $S$ and $T$ be automorphisms of a probability space and let $ S \otimes S $ be isomorphic to $T \otimes T $. Could $S$ be not isomorphic to $T$? Our note contains a simple answer to this question and a generalization of Kulaga's result on the corresponding isomorphism for some class of flows (see arXiv:1101.4975). We show that the isomorphism of weakly mixing flows $ S_t \otimes S_t $ and $ T_t \otimes T_t $ implies the isomorphism of the flows $S_t$ and $T_t$, if the latter has an integral weak limit.