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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
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Abstract

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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