Standardness of sequences of σ-fields given by certain endomorphisms
Volume 157 / 1998
Abstract
Let E be an ergodic endomorphism of the Lebesgue probability space {X, ℱ, μ}. It gives rise to a decreasing sequence of σ-fields $ℱ, E^{-1}ℱ, E^{-2}ℱ,...$ A central example is the one-sided shift σ on $X = {0, 1}^ℕ$ with $\frac 12,\frac 12$ 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 $(x, y) → (σ(x), T^{x(1)}(y))$. 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.
