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1998 | 157 | 2-3 | 175-189
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Standardness of sequences of σ-fields given by certain endomorphisms

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 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.
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  • Department of Mathematics, University of California at Berkeley Berkeley, California 94720, U.S.A.
  • Department of Mathematics, University of Maryland College Park, Maryland 20742, U.S.A.
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