Let τ and σ be two commuting ergodic measure preserving transformations of a probablity space, and Cob(τ) ,Cob(σ) be the sets of their coboundaries. We show that the inclusion Cob(σ) is contained in Cob(τ) holds if and only ifσ =τ~n for some n implied by Z. The transformations τ and σ have exactly the same coboundaries if and only if sigma = τ~+-1. Some related results and open questions are discussed.
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