×

zbMATH — the first resource for mathematics

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

MSC:
28D05 Measure-preserving transformations
28D20 Entropy and other invariants
47A35 Ergodic theory of linear operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.