Joint ergodicity for commuting transformations and applications to polynomial sequences

• 成果类型: Article
• 署名作者: Frantzikinakis, Nikos; Kuca, Borys
• 署名单位: University of Crete; Jagiellonian University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01313-w
• 发表日期: 2025
• 页码: 621-706
• 关键词: norm convergence; multiple recurrence; szemeredi theorem; averages

摘要

We give necessary and sufficient conditions for joint ergodicity results of collections of sequences with respect to systems of commuting measure preserving transformations. Combining these results with a new technique that we call seminorm smoothing, we settle several conjectures related to multiple ergodic averages of commuting transformations with polynomial iterates. We show that the Host-Kra factor is characteristic for pairwise independent polynomials, and that under certain ergodicity conditions the associated ergodic averages converge to the product of integrals. Moreover, when the polynomials are linearly independent, we show that the rational Kronecker factor is characteristic and deduce Khintchine-type lower bounds for the related multiple recurrence problem. Finally, we prove a nil plus null decomposition result for multiple correlation sequences of commuting transformations in the case where the iterates are given by families of pairwise independent polynomials.