Journal d’Analyse Mathématique

, Volume 51, Issue 1, pp 182–227 | Cite as

Joining-rank and the structure of finite rank mixing transformations

  • Jonathan L. King


A new isomorphism invariant of zero-entropy maps, calledjoiningrank, is presented. Written jrk(T), it is a value in N ∪ {∞}. The depth of factors ofT, and the size of its essential commutant EC(T), are upper bounded by jrk(T). IfT is mixing then jrk(T)≦rank(T). ForT with finite joining rank, we obtain a structure theorem for the commutant group ofT: it is a certain twisted product of Z with EC(T). As forT itself, it must be anm-point extension of thenth power of a prime transformationS having trivial commutant. Also, jrk(T)=m·n·jrk(S).

Thecovering-number, k(T), is a number in [0, 1] obeying 1/k(T)≦rank(T). Letting α(T)∈[0, 1] denoteT’s partial mixing, jrk(T) is dominated by 1/[k(T)+α(T)-1]. In particular, a rank-1T with partial mixing exceeding 1/2 has minimal self-joinings.

Combined with Kalikow’s deep theorem that, forT rank one, 2-fold mixing implies mixing of all orders, our technique yields that a mixing suchT has minimal self-joinings of all orders. ThusT may be used as the seed for Rudolph's counterexample machine.


Exact Sequence Semidirect Product Group Extension Finite Rank Finite Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    M. Akcoglu and R. Chacón,Approximation of commuting transformations, Proc. Am. Math. Soc.32 (1972), 111–119.MATHCrossRefGoogle Scholar
  2. 2.
    B. V. Chacón,Approximation and spectral multiplicity, inContributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970), Springer, Berlin, 1970, pp. 18–27.CrossRefGoogle Scholar
  3. 3.
    A. del Junco and D. Rudolph,On ergodic actions whose self-joinings are graphs, Ergodic Theory and Dynamical Systems, to appear.Google Scholar
  4. 4.
    A. del Junco, A. M. Rahe and L. Swanson,Chacón's automorphism has minimal self-joinings, J. Analyse Math.37 (1980), 276–284.MATHMathSciNetGoogle Scholar
  5. 5.
    S. Ferenczi,Systemes localement de rang un, Ann. Inst. Henri Poincaré20, (1984), 35–51.MATHMathSciNetGoogle Scholar
  6. 6.
    N. Friedman,Partially mixing of all orders and factors, preprint.Google Scholar
  7. 7.
    N. A. Friedman and D. S. Ornstein,On partially mixing transformations, Indiana Univ. Math. J.20 (1970), 767–775.CrossRefMathSciNetGoogle Scholar
  8. 8.
    N. A. Friedman, P. Gabriel and J. L. King,An invariant for rank-1rigid transformations, Ergodic Theory and Dynamical Systems (1988), to appear.Google Scholar
  9. 9.
    H. Furstenberg,Disjointedness in ergodic theory, minimal sets, and a problem in diaphantine approximation, Math. Syst. Theory,1 (1967), 1–49.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    H. Furstenberg and B. Weiss,The infinite multipliers of infinite ergodic transformations, Lecture Notes in Mathematics #668, Springer-Verlag, Berlin, 1977, pp. 127–132.Google Scholar
  11. 11.
    S. Kalikow,Twofold mixing implies threefold mixing for rank one transformations, Ergodic Theory and Dynamical Systems4 (1984), 237–259.MATHMathSciNetGoogle Scholar
  12. 12.
    J. L. King,The commutant is the weak closure of the powers, for rank-1transformations, Ergodic Theory and Dynamical Systems6 (1986), 363–384.MATHMathSciNetGoogle Scholar
  13. 13.
    J. L. King,For mixing transformations rank(T *)=k·rank(T), Isr. J. Math.56 (1986), 102–122.MATHCrossRefGoogle Scholar
  14. 14.
    J. L. King,A lower bound on the rank of mixing extensions, Isr. J. Math.59 (1987), 377–380.MATHCrossRefGoogle Scholar
  15. 15.
    D. Newton,Coalescence and spectrum of automorphisms of a Lebesgue space, Z. Wahrscheinlichkeitstheor. Verw. Geb.19 (1971), 117–122.CrossRefGoogle Scholar
  16. 16.
    D. S. Ornstein,On the root problem in ergodic theory, inProc. of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Univ. of California Press, 1970, pp. 347–356.Google Scholar
  17. 17.
    D. Rudolph,An example of a measure-preserving map with minimal self-joinings, and applications, J. Analyse Math.35 (1979), 97–122.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 1988

Authors and Affiliations

  • Jonathan L. King
    • 1
  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA

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