×

zbMATH — the first resource for mathematics

Abelian cocycles for nonsingular ergodic transformations and the genericity of type \(III_ 1\) transformations. (English) Zbl 0623.58010
The authors prove that in the space of nonsingular transformations of a Lebesgue probability space the type \(III_ 1\) ergodic transformations form a dense \(G_{\delta}\) set with respect to the coarse topology. They also prove that for any locally compact second countable abelian group H, and any ergodic type III transformation T, it is generic in the space of H-valued cocycles for the integer action given by T that the skew product of T with the cocycle is orbit equivalent to T. Similar results are given for ergodic measure-preserving transformations as well.

MSC:
37A99 Ergodic theory
PDF BibTeX Cite
Full Text: DOI EuDML
References:
[1] [CK]Choksi, J., Kakutani, S.: Residuality of ergodic measurable transformations and of ergodic transformations which preserve an infinite measure. Indiana Univ. Math. J.28, 453-469 (1979). · Zbl 0401.28017
[2] [GS1]Golodets, V., Sinelshchikov, S.: Locally compact groups appearing as ranges of cocycles of ergodic ?-actions. Ergod. Th. Dyn. Sys.5, 47-57 (1985). · Zbl 0574.22005
[3] [GS2]Golodets, V., Sinelshchikov, S.: Existence and uniqueness of cocycles of an ergodic automorphism with dense ranges in amenable groups. Preprint. (1983).
[4] [H1]Halmos, P.: Lectures on ergodic theory. Publ. Math. Soc. of Japan 3. Tokyo. 1956. · Zbl 0073.09302
[5] [HO]Hamachi, T., Osikawa, M.: Ergodic groups of automorphisms and Krieger’s theorem. Sem Math. Sci.3 (1981). · Zbl 0472.28015
[6] [H2]Hawkins, J.: Topological properties of type III1 diffeomorphisms. Preprint (1981).
[7] [H3]Herman, M.: Construction de difféomorphismes ergodiques. Preprint. (1979).
[8] [I]Ionescu Tulcea, A.: On the category of certain classes of transformations in ergodic theory. Trans. Amer. Math. Soc.114, 261-279 (1965). · Zbl 0178.38501
[9] [K1]Katznelson, Y.: Sigma-finite invariant measures for smooth mappings of the circle. J. d’Analyse Math.31, 1-18 (1977). · Zbl 0346.28012
[10] [K2]Krieger, W.: On the Araki-Woods asymptotic ratio set and non-singular transformations of a measure space. Contributions to ergodic theory and probability, Lect. Notes Math.160, 158-177 (1970).
[11] [K3]Krieger, W.: On ergodic flows and isomorphism of factors. Math. Ann.223, 18-70 (1976). · Zbl 0332.46045
[12] [M1]Mackey, G.: Ergodic theory and virtual groups. Math. Ann.166 187-207 (1966). · Zbl 0178.38802
[13] [M2]Maharam, D.: Incompressible transformations. Fund. Math.56, 35-50 (1964). · Zbl 0133.00304
[14] [PS]Parthasarathy, K. R., Schmidt, K.: On the cohomology of a hyperfinite action. Mh. Math.84, 37-48 (1977). · Zbl 0384.28017
[15] [S1]Sachdeva, U.: On category of mixing in infinite measure spaces. Math. Systems. Th.5, 319-330 (1971). · Zbl 0226.28008
[16] [S2]Schmidt, K.: Lectures on cocycles of ergodic transformation groups. Macmillan Lectures in Math., 1, Macmillan Co. of India, New Delhi, (1977).
[17] [Z1]Zimmer, R.: Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Funct. Anal.27, 350-372 (1978). · Zbl 0391.28011
[18] [Z2]Zimmer, R.: Random walks on compact groups and the existence of cocycles. Isr. J. Math.26, 84-90 (1977). · Zbl 0344.28010
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.