Abstract
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.
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Research partially supported by NSF grant DMS 8501519.
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King, J.L. Joining-rank and the structure of finite rank mixing transformations. J. Anal. Math. 51, 182–227 (1988). https://doi.org/10.1007/BF02791123
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DOI: https://doi.org/10.1007/BF02791123