×

Extensions of cocycles for hyperfinite actions and applications. (English) Zbl 0887.28008

Let \(T: (X,{\mathcal B},\mu)\to (X,{\mathcal B},\mu)\) be an ergodic automorphism on a standard Borel probability space, and \(G\) a compact metric group. A measurable transformation \(\phi: X\to G\) (called a cocycle) gives rise to the skew product transformation \(T_\phi: X\times G\to X\times G\) defined by \[ T_\phi(x, g)= (Tx, \phi(x)g), \] again an automorphism of the product space (using Haar measure on \(G\)).
Suppose \(S\in C(T)\), the centralizer of \(T\), then \(S\) “lifts” to the centralizer of \(T_\phi\) if there is a measurable solution \(f: X\to G\) to the equation \[ f(Tx)\phi(Sx)f(x)^{-1}= \phi(x). \] It turns out that many questions regarding skew products reduce to questions about this functional equation. In an earlier paper [Ergodic Theory Dyn. Syst. 12, No. 4, 769-789 (1992; Zbl 0785.58030)], M. Lemańczyk, P. Liardet and J.-P. Thouvenot approached this type of problem by considering the \(\mathbb{Z}^2\)-action \((S,T)\), defined by the commuting pair \(S\) and \(T\). The authors remark that this expresses the fact that the \(\mathbb{Z}\)-cocycle \(\phi:X\to G\) for \(T\) has been extended to a cocycle for the \(\mathbb{Z}^2\)-action \((S,T)\). This leads the authors to the study of the problem of extending a given cocycle for an action of a subgroup of some group of automorphisms to a cocycle of the whole group. Specifically, let \({\mathcal R}\) be a countable group which acts freely and ergodically on \((X,{\mathcal B},\mu)\), and let \(\mathcal H\) be a subgroup which also acts ergodically, then the authors answer questions of the following type:
1) Is the set of \({\mathcal H}\)-cocycles which have extensions to \({\mathcal R}\)-cocycles of first category in the set of \({\mathcal H}\)-cocycles?
2) Is the set of those \({\mathcal R}\)-cocycles whose restrictions to \({\mathcal H}\) are ergodic, residual in the set of \(\mathcal H\)-cocycles?

MSC:

28D15 General groups of measure-preserving transformations
28D05 Measure-preserving transformations

Citations:

Zbl 0785.58030
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Aaronson, J., Lemańczyk, M., Volny, D.: A salad of cocyles (preprint). · Zbl 0966.28010
[2] Christensen, J. P. R.: Topology and Borel Structures. Amsterdam: North Holland. 1975.
[3] Connes, A., Feldman, J., Weiss, B.: An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys.1, 431–450 (1981). · Zbl 0491.28018
[4] Cornfeld, I. P., Fomin, S. V., Sinai, Ya. G. Ergodic Theory. Berlin-Heidelberg-New York: 1982.
[5] Connes, A., Krieger, W.: Measure space automorphisms, the normalizers of their full groups, and approximate finiteness. J. Funct. Anal.24, 336–352 (1977). · Zbl 0369.28013 · doi:10.1016/0022-1236(77)90062-3
[6] Del Junco, A., Lemańczyk, M., Mentzen, M. K. Semisimplicity, joinings and group extensions. Studia Math.112, 141–164 (1995). · Zbl 0814.28007
[7] Del Junco, A., Rudolph, D. J.: On ergodic actions whose self-joinings are graphs. Ergod. Th. & Dynam. Sys.7, 531–557 (1987). · Zbl 0646.60010
[8] Dye, H. A.: On groups of measure preserving transformations I. Amer. J. Math.81, 119–159 (1959). · Zbl 0087.11501 · doi:10.2307/2372852
[9] Feldman, J., Moore, C. C.: Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Amer. Math. Soc.234, 289–324 (1977). · Zbl 0369.22009 · doi:10.1090/S0002-9947-1977-0578656-4
[10] Ferenczj, S., Lemańczyk, M.: Rank is not a spectral invariant. Studia Math.,98, 227–230 (1991). · Zbl 0728.28014
[11] Glasner, E.: For Bernoulli transformations the smallest family of natural factors comprises all factors. Ergodic Theory and its Connections with Harmonic Analysis. London Math. Soc. 1995. · Zbl 0967.37006
[12] Golodets, V. Ya., Danilenko, A. I., Bezuglyi, S. I.: On cocycles of ergodic dynamical system and automorphisms compatible with them. Adv. in Soviet Math.19, 73–96 (1994). · Zbl 0872.28008
[13] Golodets, V. Ya, Sinelshchikov, S.D.: Locally compact groups appearing as ranges of cocycles of ergodic \(\mathbb{Z}\)-actions. Ergod. Th. & Dynam. Sys.5, 47–57 (1985). · Zbl 0574.22005
[14] Goodson, G. R., del Junco, A., Lemańczyk, M., Rudolph, D.: Ergodic Transformations conjugate to their inverses by involutions. Ergod. Th. & Dynam. Sys.16, 97–124 (1996). · Zbl 0846.28008
[15] Iwasawa, K.: On some types of topological groups. Ann. of Math.50, 507–558 (1949). · Zbl 0034.01803 · doi:10.2307/1969548
[16] King, J.: The commutant is the weak closure of the powers, for rank-1 trans-formations. Ergod. Th. & Dynam. Sys.6, 363–384 (1986). · Zbl 0595.47005
[17] Kunugui, K.: Sur un problème de M.E. Szpilrajn. Proc. Japan Acad. Ser. A Math. Sci.16 70–108 (1940). · JFM 66.0360.01
[18] Kwiatkowski, J.: Factors of ergodic group extensions of rotations. Studia Math.103, 123–131 (1992). · Zbl 0809.28014
[19] Larman, D. G.: Projecting and uniformising borel sets withK {\(\sigma\)}-sections I. Mathematika19, 231–244 (1972). · Zbl 0263.54032 · doi:10.1112/S0025579300005696
[20] Larman, D. G.: Projecting and uniformising Borel sets withK {\(\sigma\)}-sections II. Mathematika,20, 233–246 (1973). · Zbl 0287.54040 · doi:10.1112/S0025579300004848
[21] Lemanczyk, M., Liardet, P., Thouvenot, J.-P.: Coalescence of circle extensions of measurepreserving transformations. Ergod. Th. & Dynam. Sys.12, 769–789 (1992). · Zbl 0785.58030
[22] Mentzen, M. K.: Ergodic properties of group extensions of dynamical systems with discrete spectra. Studia Math.101, 19–31 (1991). · Zbl 0809.28015
[23] Rudolph, D. J.: An isomorphism theory for Bernoulli freeZ-skew-compact group actions. Adv. in Math.47, 241–257 (1983). · Zbl 0547.28013 · doi:10.1016/0001-8708(83)90074-9
[24] Rychlik, M.: The Wiener lemma and cocycles Proc. Amer. Math. Soc.104, 932–933 (1988). · Zbl 0687.58018 · doi:10.1090/S0002-9939-1988-0964876-6
[25] Schmidt, K.: Cocycles on ergodic transformation groups. Delhi: MacMillan (India), 1977. · Zbl 0421.28017
[26] Veech, W. A.: Strict ergodicity in zero dimensional systems and the Kronecker-Weyl theorem mod 2. Trans. Amer. Math. Soc.140, 1–35 (1969). · Zbl 0201.05601
[27] Veech, W. A.: A criterion for a process to be prime. Mh. Math.94, 335–341 (1982). · Zbl 0499.28016 · doi:10.1007/BF01667386
[28] Zimmer, R. J.: Extensions of ergodic group actions. Illinois J. Math.20, 373–409 (1976). · Zbl 0334.28015
[29] Zimmer, R. J.: Ergodic theory and semisimple Lie groups. Basel-Berlin: Birkhäuser. 1984.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.