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Weakly isomorphic transformations that are not isomorphic. (English) Zbl 0628.60015
A new method for construction of transformations T: (X\({}_ i,{\mathcal B}_ i,\mu _ i)\), \(i=1,2\), that are factors of each other but that are not measure-theoretically isomorphic is provided. This method uses ergodic product cocycles of the form \(\phi \circ S^{i_ 1}\times \phi \circ S^{i_ 2}\times...\), where \(\phi\) : \(X\to Z_ 2\) is a cocycle, S belongs to the centralizer of T and T is an ergodic translation on a compact, monothetic group X.

MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
28D05 Measure-preserving transformations
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[1] Christol, G., Kamae, T., Mendès-France, M., Rauzy, G.: Suites algébriques, automates et substitutions, BSMF 108, 401–419 (1980) · Zbl 0472.10035
[2] Coven. E., Keane, M.: The structure of substitution minimal sets. TAMS 62, 89–102 (1971) · Zbl 0222.54053
[3] Feldman, J.: Borel structures and invariants for measurable transformations. PAMS 46, 383–394 (1974) · Zbl 0292.28008
[4] Fürstenberg, H.: Disjointness in ergodic theory, minimal sets and Diophantine approximation. Math. Syst. Theory 1, 1–49 (1967) · Zbl 0146.28502
[5] Del Junco, A., Rahe, A. M., Swanson, L.: Chacon transformation has minimal self-joinings. J. Anal. Math. 37, 276–284 (1980) · Zbl 0445.28014
[6] Del Junco, A., Rudolph, D.: On ergodic actions whose self-joinings are graphs. To appear in Ergodic Theory Dyn. Syst. · Zbl 0646.60010
[7] Keane, M.: Generalized Morse sequences. Z. Wahrscheinlichkeitstheor. Verw. Geb. 10, 335–353 (1968) · Zbl 0162.07201
[8] Kwiatkowski, J.: Isomorphism of regular Morse dynamical systems. Studia Math. 62, 59–89 (1982) · Zbl 0525.28018
[9] Kwiatkowski, J., Lemańczyk, M.: Centralizer of group extensions, preprint · Zbl 0857.28012
[10] Lemańczyk, M.: The centralizer of Morse shifts. Ann. Univ. Clermont-Ferrand 87, 43–56 (1985) · Zbl 0587.28016
[11] Lemańczyk, M.: Toeplitz Z 2-extensions. To appear in Ann. Henri Poincaré Inst. 24, 1–43 (1988) · Zbl 0647.28013
[12] Lemańczyk, M.: Factors of coalescent automorphisms. To appear in Studia Math. · Zbl 0695.28005
[13] Lemańczyk, M., Mentzen, M. K.: Metric properties of substitutions. Composito Math. 65, 241–263 (1988) · Zbl 0696.28009
[14] Newton, D.: Coalescence and spectrum of automorphisms of a Lebesgue space. Z. Wahrscheinlichkeitstheor. Verw. Geb. 19, 117–122 (1971) · Zbl 0209.36302
[15] Newton, D.: On canonical factors of ergodic dynamical systems. J. London Math. Soc. (2) 19, 129–136 (1979) · Zbl 0425.28012
[16] Ornstein, D. Rudolph, D., Weiss, B.: Equivalence of measure-preserving transformations. Memoirs AMS 37, 262 (1982) · Zbl 0504.28019
[17] Parry, W.: Compact abelian group extensions of discrete dynamical systems. Z. Wahrscheinlichkeitstheor. Verw. Geb. 13, 95–113 (1969) · Zbl 0184.26901
[18] Polit, S.: Weakly isomorphic maps need not be isomorphic. Ph. D. dissertation, Stanford 1974
[19] Rudolph, D.: An example of a measure-preserving map with minimal self-joinings and applications. J. Anal. Math. 35, 97–122 (1979) · Zbl 0446.28018
[20] Snai, Y.G.: On weak isomorphism of transformations with invariant measure (in Russian). Math. Sb. 63, 23–42 (1963)
[21] Thouvenot, J.P.: The metrical structure of some Gaussian processes. Proc. Erg. Theory Rel.; Topics II, pp. 195–198, Georgenthal 1986
[22] Veech, W.A.: A criterion for a process to be prime. Monatsh. Math. 94, 335–341 (1982) · Zbl 0499.28016
[23] Zimmer, R.: Extensions of ergodic actions. Illinois J. Math. 20, 373–409 (1976) LH · Zbl 0334.28015
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