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Joining-rank and the structure of finite rank mixing transformations. (English) Zbl 0665.28010
This is an extensive paper presenting a new isomorphism invariant of ergodic maps called joining-rank (jrk). Its definition and some fundamental properties may be found in Section 2. In particular, jrk(T) dominates the size \(| EC(T)|\) of the essential commutant of T; the finiteness of jrk(T) forces T to have zero entropy. The main result of Section 3 shows that a map with sufficiently large covering number and partial mixing number has finite joining-rank. For a mixing transformation T we have the inequality \(jrk(T)\leq rk(T).\) Section 4 provides a theorem which says that if \(jrk(T)<\infty\) then T is a finite extension of a power of a prime transformation with trivial commutant. The last part of the paper gives an algebraic structure theorem for the commutant group of T with finite joining-rank. The introduction and Section 1 provide some elementary facts concerning the numerous notions used in the article.
Reviewer: W.Jarczyk

28D05 Measure-preserving transformations
28D20 Entropy and other invariants
47A35 Ergodic theory of linear operators
Full Text: DOI
[1] M. Akcoglu and R. Chacón,Approximation of commuting transformations, Proc. Am. Math. Soc.32 (1972), 111–119. · Zbl 0229.28010 · doi:10.1090/S0002-9939-1972-0289745-7
[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.
[3] A. del Junco and D. Rudolph,On ergodic actions whose self-joinings are graphs, Ergodic Theory and Dynamical Systems, to appear. · Zbl 0646.60010
[4] A. del Junco, A. M. Rahe and L. Swanson,Chacón’s automorphism has minimal self-joinings, J. Analyse Math.37 (1980), 276–284. · Zbl 0445.28014 · doi:10.1007/BF02797688
[5] S. Ferenczi,Systemes localement de rang un, Ann. Inst. Henri Poincaré20, (1984), 35–51. · Zbl 0535.28010
[6] N. Friedman,Partially mixing of all orders and factors, preprint.
[7] N. A. Friedman and D. S. Ornstein,On partially mixing transformations, Indiana Univ. Math. J.20 (1970), 767–775. · Zbl 0213.07504 · doi:10.1512/iumj.1971.20.20061
[8] N. A. Friedman, P. Gabriel and J. L. King,An invariant for rank-1rigid transformations, Ergodic Theory and Dynamical Systems (1988), to appear. · Zbl 0621.28011
[9] H. Furstenberg,Disjointedness in ergodic theory, minimal sets, and a problem in diaphantine approximation, Math. Syst. Theory,1 (1967), 1–49. · Zbl 0146.28502 · doi:10.1007/BF01692494
[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. · Zbl 0385.28009
[11] S. Kalikow,Twofold mixing implies threefold mixing for rank one transformations, Ergodic Theory and Dynamical Systems4 (1984), 237–259. · Zbl 0552.28016 · doi:10.1017/S014338570000242X
[12] J. L. King,The commutant is the weak closure of the powers, for rank-1transformations, Ergodic Theory and Dynamical Systems6 (1986), 363–384. · Zbl 0595.47005 · doi:10.1017/S0143385700003552
[13] J. L. King,For mixing transformations rank(T * )=k\(\cdot\)rank(T), Isr. J. Math.56 (1986), 102–122. · Zbl 0626.47012 · doi:10.1007/BF02776244
[14] J. L. King,A lower bound on the rank of mixing extensions, Isr. J. Math.59 (1987), 377–380. · Zbl 0647.28012 · doi:10.1007/BF02774146
[15] D. Newton,Coalescence and spectrum of automorphisms of a Lebesgue space, Z. Wahrscheinlichkeitstheor. Verw. Geb.19 (1971), 117–122. · Zbl 0209.36302 · doi:10.1007/BF00536902
[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.
[17] D. Rudolph,An example of a measure-preserving map with minimal self-joinings, and applications, J. Analyse Math.35 (1979), 97–122. · Zbl 0446.28018 · doi:10.1007/BF02791063
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