Belinskaya’s theorem is optimal

Belinskaya’s theorem states that given an ergodic measure-preserving transformation, any other transformation with the same orbits and an L$^1$ cocycle must be flip-conjugate to it. Our main result shows that this theorem is optimal: for all $p \lt 1$ the integrability condition on the cocycle cannot be relaxed to being in L$^p$. This also allows us to answer a question of Kerr and Li: for ergodic measure-preserving transformations, Shannon orbit equivalence does not boil down to flip-conjugacy.