Belinskaya's theorem is optimal

TL;DR

This paper proves that Belinskaya's theorem's integrability condition is optimal and demonstrates that Shannon orbit equivalence is strictly broader than flip-conjugacy for ergodic measure-preserving transformations.

Contribution

It establishes the sharpness of Belinskaya's theorem by showing the condition cannot be weakened below L^1 and clarifies the relationship between Shannon orbit equivalence and flip-conjugacy.

Findings

Belinskaya's theorem is optimal at L^1 integrability.

Shannon orbit equivalence is not equivalent to flip-conjugacy.

The integrability condition cannot be relaxed to L^p for p<1.

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.