Hostname: page-component-6766d58669-tq7bh Total loading time: 0 Render date: 2026-05-20T02:14:59.474Z Has data issue: false hasContentIssue false
  • English
  • Français

Product and Markov measures of type III

Part of: Measure-theoretic ergodic theory

Published online by Cambridge University Press:  09 April 2009

Anthony H. Dooley ,
Ivo Klemeš  and
Anthony N. Quas
Anthony H. Dooley
Affiliation:
School of Mathematics, University of New South Wales, Sydney 2052, Australia e-mail: a.dooley@unsw.edu.au
Ivo Klemeš
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. W., Montreal, Quebec, Canada, H3A 2K6 e-mail: klemes@math.mcgill.ca
Anthony N. Quas
Affiliation:
Department of Mathematical Sciences, University of Memphis Memphis, TN 38152USA e-mail:quasa@msci.memphis.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We give some explicit constructions of type III product measures with various properties, resolving some conjectures of Brown, Dooley and Lake. We also define a family of Markov odometers of type III0 and show that the associated flow is approximately transitive.

Keywords

AT propertyG-measuresorbit equivalencenon-singular transformations.

MSC classification

Secondary: 28D15: General groups of measure-preserving transformations

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 1 , August 1998 , pp. 84 - 110
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Brown, G. and Dooley, A. H., ‘Ergodic measures are of weak product type’, Math. Proc. Cambridge Philos. Soc. 98 (1985), 129–145.CrossRefGoogle Scholar
[2]Brown, G. and Dooley, A. H., ‘Odometer actions on G-measures’, Ergodic Theory Dynamical Systems 11 (1991), 279–307.CrossRefGoogle Scholar
[3]Brown, G., Dooley, A. H. and Lake, J., ‘On the Krieger-Araki-Woods ratio set’, Tôhoku Math. J. 47 (1995), 1–13.CrossRefGoogle Scholar
[4]Choksi, J. R., Hawkins, J. M. and Prasad, V. S., ‘Abelian cocycles for nonsingular ergodic transformations and the genericity of type III1 transformations’, Monatsh. Math. 103 (1987), 187–205.CrossRefGoogle Scholar
[5]Connes, A. and Woods, E. J., ‘Approximately transitive flows and ITPFI factors’, Ergodic Theory Dynamical Systems 5 (1985), 203–236.CrossRefGoogle Scholar
[6]Chung, K. L.A course in probability theory (Academic Press, New York, 1974).Google Scholar
[7]Dooley, A. H. and Hamachi, T., ‘Markov odometer actions not of product type’, Trans. Amer Math. Soc., to appear.Google Scholar
[8]Dye, H., ‘On groups of measure preserving transformations II’, Amer J. Math. 95 (1963), 551–576.CrossRefGoogle Scholar
[9]Hamachi, T. and Osikawa, M., ‘Computation of the associated flows of IT P FI 2 factors of type I I I0’, in: Geometric methods in operator algebras (eds. Araki, H. and Effros, E. G.) Pitman Research Notes 123 (Longman, New York, 1986), pp. 126–210.Google Scholar
[10]Hamachi, T., ‘A measure-theoretical proof of the Connes-Woods theorem on AT-flows’, Pacific J. Math. 154(1992), 67–85.CrossRefGoogle Scholar
[11]Hamachi, T., Oka, Y. and Osikawa, M., ‘A classification of ergodic non-singular transformation groups’, Mem. Fac. Sci., Kyushu Univ. Ser A 28(1974), 115–133.Google Scholar
[12]Hawkins, J. M., ‘Properties of ergodic flows associated to product odometers’, Pacific J. Math. 141 (1990), 287–294.CrossRefGoogle Scholar
[13]Krieger, W., ‘On non-singular transformations of a measure space I’, Z Wahrscheinlichkeitstheorie verw. Geb. 11 (1969), 83–91;CrossRefGoogle Scholar
II, Z Wahrscheinlichkeitstheorie verw. Geb. 11 (1969), 98–119.CrossRefGoogle Scholar
[14]Krieger, W., ‘On the Araki-Woods asymptotic ratio set and non-singular transformations of a measure space’, Lecture Notes in Math. 160 (Springer, Berlin, 1970), pp. 158–177.Google Scholar
[15]Krieger, W., ‘On the infinite product construction of non-singular transformations of measure space’, Invent. Math. 15 (1972), 144–163.CrossRefGoogle Scholar
[16]Moore, C. C., ‘Invariant measures on product spaces’, Proceedings of the fifth Berkeley symposium, (1967), 447–459.Google Scholar
[17]Sutherland, C. E., Notes on orbit equivalence, Lecture note series 23 (Univ. i Oslo, 1976).Google Scholar