The second centralizer of a Bernoulli shift is just its powers |
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| Authors: | Daniel J Rudolph |
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| Institution: | (1) Department of Mathematics, University of California, 94720 Berkeley, Calif, USA |
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| 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
1
i
°T
2
i
|i, j ∈ Z}, whereT
1 is a Bernoulli shift of arbitrary entropy, but for anyj ≠ 0,C({T
1,T
2
i}
={T
1
i
°T
2
k
l, k ∈ Z}. The construction is a two-dimensional analogue of Ornstein’s “rank one mixing” transformation. |
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| Keywords: | |
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