The self-joinings of rank two mixing transformations

Ullman, Daniel

doi:10.1090/S0002-9939-1992-1076581-X

The self-joinings of rank two mixing transformations
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Proc. Amer. Math. Soc. 114 (1992), 53-60 Request permission

Abstract:

The class of rank one mixing transformations has been known for some time to have certain so-called "exotic" properties. Specifically, any rank one mixing $T$ has only the two trivial factors, and nothing can commute with $T$ but powers of $T$ itself. This much was known to Ornstein in 1969 [3]. In 1983 J. King showed that rank one mixing transformations possess an even stronger property known as minimal self-joinings (MSJ). In this note we investigate how these results can be extended to the case of rank two mixing transformations. In this class, it is possible for there to exist nontrivial factors and commuting transformations: the square of a rank one mixing transformation and certain two point extensions of a rank one mixing transformation are rank two mixing [2]. What we prove is that, other than those two kinds of rank two mixing transformations, this class also has MSJ.

References

J. R. Baxter, A class of ergodic transformations having simple spectrum, Proc. Amer. Math. Soc. 27 (1971), 275–279. MR 276440, DOI 10.1090/S0002-9939-1971-0276440-2
Jonathan King, For mixing transformations $\textrm {rank}(T^k)=k\cdot \textrm {rank}(T)$, Israel J. Math. 56 (1986), no. 1, 102–122. MR 879918, DOI 10.1007/BF02776244
Donald S. Ornstein, On the root problem in ergodic theory, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 347–356. MR 0399415
Daniel J. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. Analyse Math. 35 (1979), 97–122. MR 555301, DOI 10.1007/BF02791063