Abstract
A new isomorphism invariant of certain measure preserving flows, using sequences of integers, is introduced. Using this invariant, we are able to construct large families of type III0 systems which are not orbit equivalent. In particular we construct an uncountable family of nonsingular ergodic transformations, each having an associated flow that is approximately transitive (and therefore of zero entropy), with the property that the transformations are pairwise not orbit equivalent.
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Araki H., Woods E.J.: A classification of factors. Publ. RIMS, Kyoto Univ. 4, 51–130 (1968)
Boshernitzan, M.: Discrete “Orders of Infinity”. Am. J. Math. 106:5, pp. 1147–1198 (1984)
Connes A., Feldman J., Weiss B.: An Amenable Equivalence Relation Is Generated By a Single Transformation. Ergod. Theory Dyn. Syst. 1, 431–450 (1981)
Connes, A., Takesaki, M.: The Flow of Weights on Factors of Type III. Tohoku Math. J. (2), 29:4, 473–575 (1977)
Connes A., Woods E.J.: Approximately Transitive Flows and ITPFI Factors. Ergod. Theory Dyn. Syst. 5, 203–236 (1985)
Cornfeld I.P., Fomin S.V., Sinai Ya.G.: Ergodic Theory. Springer, Berlin (1982)
Dye H.A.: On groups of measure-preserving transformations II. Am. J. Math. 85, 551–576 (1963)
Hamachi T., Osikawa M.: Ergodic Groups of Automorphisms and Krieger’s Theorems. Sem. Math. Sci. Keio Univ. 3, 1–113 (1981)
Katznelson Y., Weiss B.: The classification of non-singular actions, revisited. Ergod. Theory Dyn. Syst. 11, 333–348 (1991)
Krieger W.: On the Araki-Woods Asymptotic Ratio Set and Non-Singular Transformations of a Measure Space, Lecture Notes in Math. 160, pp. 158–177. Springer, Berlin (1970)
Krieger W.: On a class of hyperfinite factors that arise from null-recurrent Markov chains. J. Funct. Anal. 7, 27–42 (1971)
Krieger W.: On ergodic flows and the isomorphism of factors. Math. Ann. 223, 19–70 (1976)
Osikawa, M.: Point Spectra of Non-Singular Flows. Publ. Res. Inst. Math. Sci. 13:1, 167–172 (1977)
Park K.: Even Kakutani equivalence via α-equivalences and β-equivalences. J. Math. Anal. Appl. 195, 335–353 (1995)
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Communicated by Klaus Schmidt.
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Dooley, A.H., Hawkins, J. & Ralston, D. Families of type III0 ergodic transformations in distinct orbit equivalent classes. Monatsh Math 164, 369–381 (2011). https://doi.org/10.1007/s00605-010-0258-0
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DOI: https://doi.org/10.1007/s00605-010-0258-0