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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

##### MSC:
 28D05 Measure-preserving transformations 28D20 Entropy and other invariants 47A35 Ergodic theory of linear operators
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##### References:
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