Abstract
If ℐ is a collection of measure preserving transformations of a probability space, byC(ℐ), the centralizer of ℐ, we mean the group of all measure preserving transformationsS such thatTS=ST for allT ∈ ℐ. We show here that ifT is a Bernoulli shift, thenC(C(T))={T i |i ∈ Z}. The proof is carried out by constructing an action of Z2, {T i1 °T i2 |i, j ∈ Z}, whereT 1 is a Bernoulli shift of arbitrary entropy, but for anyj ≠ 0,C({T 1,T i}2 ={T i1 °T k2 l, k ∈ Z}. The construction is a two-dimensional analogue of Ornstein’s “rank one mixing” transformation.
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References
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Rudolph, D.J. The second centralizer of a Bernoulli shift is just its powers. Israel J. Math. 29, 167–178 (1978). https://doi.org/10.1007/BF02762006
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DOI: https://doi.org/10.1007/BF02762006