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arXiv · 2201.06662

Belinskaya's theorem is optimal

Alessandro Carderi·Matthieu Joseph·François Le Maître·Romain Tessera
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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.

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Alessandro Carderi, Matthieu Joseph, François Le Maître, Romain Tessera. 2022-01-17. Belinskaya's theorem is optimal. https://doi.org/10.4064/fm266-4-2023

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