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## Fundamenta Mathematicae

1998 | 157 | 2-3 | 175-189
Tytuł artykułu

### Standardness of sequences of σ-fields given by certain endomorphisms

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
175-189
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-10-09
poprawiono
1998-02-18
Twórcy
autor
• Department of Mathematics, University of California at Berkeley Berkeley, California 94720, U.S.A.
autor
• Department of Mathematics, University of Maryland College Park, Maryland 20742, U.S.A.
Bibliografia
• [1] R. Burton, A non-Bernoulli skew product which is loosely Bernoulli, Israel J. Math. 35 (1980), 339-348.
• [2] J. Feldman, New K-automorphisms and a problem of Kakutani, ibid. 24 (1976), 16-38.
• [3] D. Heicklen, Entropy and r equivalence, Ergodic Theory Dynam. Systems, to appear.
• [4] D. Heicklen and C. Hoffman, $T, T^{-1}$ is not standard, ibid. to appear.
• [5] D. Heicklen, C. Hoffman and D. Rudolph, Entropy and dyadic equivalence of random walks on a random scenery, preprint.
• [6] C. Hoffman, A zero entropy (T,Id) endomorphism that is not standard, Ergodic Theory Dynam. Systems, to appear.
• [7] A. del Junco, Transformations with discrete spectrum are stacking transformations, Canad. J. Math. 28 (1976), 836-839.
• [8] S. Kalikow, $T, T^{-1}$ transformation is not loosely Bernoulli, Ann. of Math. 115 (1982), 393-409.
• [9] S. Kalikow, Twofold mixing implies threefold mixing for rank one transformations, Ergodic Theory Dynam. Systems 4 (1984), 237-259.
• [10] G. Kallianpur, Some ramifications of Wiener's ideas on nonlinear prediction, in: Norbert Wiener, Collected Works (P. Masani, ed.), Vol. III, MIT Press, 1981, 402-423.
• [11] G. Kallianpur and N. Wiener, 1956, Tech. report No. 1, Office of Naval Research CU-2-56 NONR-266 (39), CIRMIP project NR-047-015.
• [12] Y. Katznelson, Ergodic automorphisms of $\mathbbT^n$ are Bernoulli shifts, Israel J. Math. 10 (1971), 186-195.
• [13] P. Lévy, Théorie de l'addition des variables aléatoires, Gauthier-Villars, Paris, 1954.
• [14] D. Lind, The structure of skew products with ergodic group automorphisms, Israel J. Math. 28 (1977), 205-248.
• [15] I. Meilijsohn, Mixing properties of a class of skew products, ibid. 19 (1974), 266-270.
• [16] G. Miles and K. Thomas, Generalized torus automorphisms are Bernoullian, in: Adv. Math. Suppl. Stud. 2, Academic Press, 1978, 231-249.
• [17] D. S. Ornstein, D. Rudolph and B. Weiss, Equivalence of measure preserving transformations, Mem. Amer. Math. Soc. 262 (1982).
• [18] M. Rosenblatt, Stationary processes as shifts of functions of independent random variables, J. Math. Mech. 8 (1959), 665-681.
• [19] A. Vershik, The theory of decreasing sequences of measurable partitions, St. Petersburg Math. J. 6 (1995), 705-761.
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Bibliografia
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