Belinskaya's theorem is optimal

Abstract

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

Some more fixes in the appendix; added a reference to M. Joseph's thesis in Remark 3.4. Final final version matching the online first version in Fundamenta Mathematicae

References in corpus (1)

• Entropy, virtual Abelianness, and Shannon orbit equivalence

Cited by in corpus (2)

• Rank-one systems, flexible classes and Shannon orbit equivalence
• Quantitative orbit equivalence for $\mathbb{Z}$-odometers