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

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 Følner set argument.