Standardness of sequences of σ-fields given by certain endomorphisms

Let E be an ergodic endomorphism of the Lebesgue probability space {X, ℱ, μ}. It gives rise to a decreasing sequence of σ-fields ${\displaystyle ℱ,E^{-1}ℱ,E^{-2}ℱ,...}$ A central example is the one-sided shift σ on ${\displaystyle X={\{0,1\}}^{\mathbb{N}}}$ with ${\displaystyle \frac{12}{,}\frac{12}{}}$ product measure. Now let T be an ergodic automorphism of zero entropy on (Y, ν). The [I|T] endomorphism} is defined on (X× Y, μ× ν) by ${\displaystyle (x,y)\to (\sigma \left(x\right),T^{x\left(1\right)}\left(y\right))}$. Here ℱ is the σ-field of μ× ν-measurable sets. Each field is a two-point extension of the one beneath it. Vershik has defined as "standard'' any decreasing sequence of σ-fields isomorphic to that generated by σ. Our main results are: THEOREM 2.1. If T is rank-1 then the sequence of σ-fields given by [I|T] is standard. COROLLARY 2.2. If T is of pure point spectrum, and in particular if it is an irrational rotation of the circle, then the σ-fields generated by [I|T] are standard. COROLLARY 2.3. There exists an exact dyadic endomorphism with a finite generating partition which gives a standard sequence of σ-fields, while its natural two-sided extension is not conjugate to a Bernoulli shift.