Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-12-01T02:46:12.194Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal self-joinings for nonsingular transformations

Published online by Cambridge University Press:  19 September 2008

Daniel J. Rudolph  and
Cesar E. Silva
Daniel J. Rudolph
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742, USA
Cesar E. Silva
Affiliation:
Department of Mathematics, Williams College, Williamstown, Massachusetts, 01267, USA
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.

The notion of minimal self-joinings for conservative nonsingular actions is defined as a restriction on the nature of rational self-joinings. The need to consider rational joinings is demonstrated by showing that any two type II1 actions whose Cartesian product is ergodic have type IIIλ nonsingular joinings. Lastly, actions of all Krieger types with minimal self-joinings are constructed. Hence these actions are prime and commute only with their powers.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 4 , December 1989 , pp. 759 - 800
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[A]Aaronson, J.. The intrinsic normalising constants of transformations preserving infinite measures. J. d'Analyse Math. 49 (1987), 239–270.Google Scholar
[ALW]Aaronson, J., Lin, M. & Weiss, B.. Mixing properties of Markov operators and ergodic transformations, and ergodicity of Cartesian products. Israel J. Math. 33 (1979), 198–224.Google Scholar
[AN]Aaronson, J. & Nadkarni, M., L∞ eigenvalues and L2 spectra of non-singular transformations. Proc. London Math. Soc. 55 (3) (1987), 538–570.Google Scholar
[EHI]Eigen, S.J., Hajian, A. & Ito, Y.. Ergodic measure preserving transformations of finite type. Tokyo J. Math. 11 (2) 1988, 459–470.Google Scholar
[Fr]Friedman, N.A.. Introduction to Ergodic Theory. Van Nostrand: New York, 1970.Google Scholar
[F1]Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 1–49.Google Scholar
[F2]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press: Princeton, 1981.Google Scholar
[HK]Hajian, A. & Kakutani, S.. Example of an ergodic measure preserving transformation on an infinite measure space. Contributions to Ergodic Theory and Probability. Lecture Notes in Mathematics 160. Springer-Verlag: Berlin, 1970, 45–52.Google Scholar
[HIK]Hajian, A., Ito, Y. & Kakutani, S.. Invariant Measures and orbits of dissipative transformations. Adv. Math. 9 (1972), 52–65.Google Scholar
[H]Halmos, P.R.. An ergodic theorem. Proc. Nat. Acad. Sci. U.S.A. 32 (1946), 156–161.Google Scholar
[HO]Hamachi, T. & Osikawa, . Ergodic groups of automorphisms and Krieger's theorems. Seminar on Math. Sci. Keio University No. 3 (1981).Google Scholar
[HW]Hawkins, J. & Woods, E. J.. Approximately transitive diffeomorphisms of the circle. Proc. Amer. Math. Soc. 90 (1984), 252–262.Google Scholar
[JK]Jones, L.K. & Krengel, U.. On transformations without finite invariant measure. Adv. Math. 12 (1974), 275–295.Google Scholar
[JRS]Junco, A.del, Rahe, M. & Swanson, L.. Chacón's automorphism has minimal self-joinings. J. d'Analyse Math. 37 (1980), 276–284.Google Scholar
[JR]Junco, A.del & Rudolph, D. J.. On ergodic actions whose self-joinings are graphs. Ergod. Th. & Dynam. Sys. 7 (1987), 531–557.Google Scholar
[K]King, J.. Joining-rank and the structure of finite rank mixing transformations. J. d'Analyse Math. 51 (1988), 182–227.Google Scholar
[Kr]Krieger, W.. On the Araki-Woods asymptotic ratio set and non singular transformations of a measure space. Lecture Notes in Mathematics 160. Springer-Verlag: Berlin, 1970, 158–177.Google Scholar
[M1]Maharam, D.. Incompressible transformations. Fund. Math. LVI (1964), 35–50.Google Scholar
[M2]Maharam, D.. On the planar representation of a measurable subfield. Measure Theory Oberwolfach 1983. Lecture Notes in Mathematics 1089. Springer-Verlag: Berlin, 1984, 47–57.Google Scholar
[P]Parry, W.. Topics in Ergodic Theory. Cambridge University Press: Cambridge, 1981.Google Scholar
[R]Rudolph, D.J.. An example of a measure preserving map with minimal self-joinings, and applications. J. d'Analyse Math. 35 (1979), 97–122.Google Scholar
[S]Schmidt, K.. Lectures on cocycles of ergodic transformation groups. Macmillan Lectures in Math. 1. Macmillan Co. of India: New Delhi, 1977.Google Scholar
[W]Weiss, B.. Orbit equivalence of nonsingular actions. Monographic No. 29. L' Enseignement Mathématique (1980), 77–107.Google Scholar