The isomorphism of measure-preserving transformations with isomorphic Cartesian powers is typical (Q1358529)

The author tries to answer the following question: If \(S\) and \(T\) are measure-preserving transformations of a Lebesgue probability space \((X,\mu)\) such that \(\times_1^d S\) and \(\times_1^d T\) are metrically isomorphic, does imply this condition that \(T\) and \(S\) are isomorphic? He proves that if \(T\) is a \(\chi\)-mixing automorphism, \(0<\chi<1\), and an automorphism \(S\) satisfies the relation \[ \Phi(S\otimes\ldots\otimes S)\Phi^{-1}= T\otimes\ldots\otimes T \] where \(\Phi\) is the operator corresponding to a measure-preserving transformation of \(\times_1^d X\) with measure \(\times_1^d \mu\), then \(S\) and \(T\) are isomorphic.