Standardness of sequences of σ-fields given by certain endomorphisms

Jacob Feldman; Daniel Rudolph

Fundamenta Mathematicae (1998)

  • Volume: 157, Issue: 2-3, page 175-189
  • ISSN: 0016-2736

Abstract

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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 1 2 , 1 2 product measure. Now let T be an ergodic automorphism of zero entropy on (Y, ν). The [I|T] endomorphismis 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.

How to cite

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Feldman, Jacob, and Rudolph, Daniel. "Standardness of sequences of σ-fields given by certain endomorphisms." Fundamenta Mathematicae 157.2-3 (1998): 175-189. <http://eudml.org/doc/212284>.

@article{Feldman1998,
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 $\frac\{1\}\{2\},\frac\{1\}\{2\}$ product measure. Now let T be an ergodic automorphism of zero entropy on (Y, ν). The [I|T] endomorphismis 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. },
author = {Feldman, Jacob, Rudolph, Daniel},
journal = {Fundamenta Mathematicae},
keywords = {Vershik's standardness; sequence of -fields; ergodic endomorphism; rank-1; pure point spectrum; dyadic endomorphism; Bernoulli shift},
language = {eng},
number = {2-3},
pages = {175-189},
title = {Standardness of sequences of σ-fields given by certain endomorphisms},
url = {http://eudml.org/doc/212284},
volume = {157},
year = {1998},
}

TY - JOUR
AU - Feldman, Jacob
AU - Rudolph, Daniel
TI - Standardness of sequences of σ-fields given by certain endomorphisms
JO - Fundamenta Mathematicae
PY - 1998
VL - 157
IS - 2-3
SP - 175
EP - 189
AB -  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 $\frac{1}{2},\frac{1}{2}$ product measure. Now let T be an ergodic automorphism of zero entropy on (Y, ν). The [I|T] endomorphismis 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.
LA - eng
KW - Vershik's standardness; sequence of -fields; ergodic endomorphism; rank-1; pure point spectrum; dyadic endomorphism; Bernoulli shift
UR - http://eudml.org/doc/212284
ER -

References

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