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arXiv · 2607.11549

Positive Rokhlin Entropy Implies Infinite $L^1$-Orbit Multiplicity: A Negative Answer to Thouvenot's Question

Zongrui Hu·Leiye Xu·Shuhao Zhang
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Abstract

We prove that every free ergodic measure-preserving action of a countably infinite amenable group with positive Rokhlin entropy has infinite complex $L^1$-orbit multiplicity, both on $L^1$ and on its mean-zero subspace $L^1_0$. This gives a negative answer to a question of J.-P. Thouvenot recorded by Iwanik and establishes the corresponding endpoint statement at $p=1$ of Iwanik's theorem that positive entropy implies infinite $L^p$-multiplicity for every $p>1$. The proof combines Malykhin's rigidity theorem for independent random variables, Seward's Bernoulli factor theorem, and a F{\o}lner set argument.

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Zongrui Hu, Leiye Xu, Shuhao Zhang. 2026-07-13. Positive Rokhlin Entropy Implies Infinite $L^1$-Orbit Multiplicity: A Negative Answer to Thouvenot's Question. https://arxiv.org/abs/2607.11549

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