Abstract
Let (X, μ) be a probability space, G a countable amenable group, and (F n ) n a left Følner sequence in G. This paper analyzes the non-conventional ergodic averages
associated to a commuting tuple of μ-preserving actions \({T_1}, \ldots {T_d}:G \curvearrowright X\) and f 1,..., f d ∈ L ∞(μ). We prove that these averages always converge in \({\left\| \cdot \right\|_2}\), and that they witness a multiple recurrence phenomenon when f 1 =... = f d = 1 A for a non-negligible set A ⊆ X. This proves a conjecture of Bergelson, McCutcheon and Zhang. The proof relies on an adaptation from earlier works of the machinery of sated extensions.
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Research supported by a fellowship from the Clay Mathematics Institute.
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Austin, T. Non-conventional ergodic averages for several commuting actions of an amenable group. JAMA 130, 243–274 (2016). https://doi.org/10.1007/s11854-016-0036-6
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DOI: https://doi.org/10.1007/s11854-016-0036-6