Coboundaries for commuting transformations (Q1305185)

The author gives necessary and sufficient conditions for two commuting ergodic measure preserving transformations on a non-atomic probability space to have the same set of coboundaries. Namely, he shows that such transformations have the same set of coboundaries if and only if the two transformations are the same or one is the inverse of the other. He also gives a continuous version of this result and shows that the condition of commuting (in the measurable case) cannot be dropped. His proofs are based on a slight modification (a non-free version) of the \(\mathbb{Z}^2\)-Rokhlin lemma.