On the isomorphism of tensor powers of ergodic flows

TL;DR

This paper addresses a fundamental question in ergodic theory about whether the isomorphism of tensor powers of flows implies the isomorphism of the flows themselves, providing a positive answer under certain conditions.

Contribution

It offers a simple proof that tensor power isomorphism implies flow isomorphism for flows with an integral weak limit, extending previous results.

Findings

Tensor power isomorphism implies flow isomorphism for flows with an integral weak limit.

Generalizes Kulaga's result to a broader class of flows.

Provides a positive answer to a longstanding question in ergodic theory.

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.