2025/06/25 by Sebastián Donoso, Andreas Koutsogiannis, Donoso, Sebastián +7
Mathematics · #28D05 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Approximation and Integration #Primary: 37A44 #Secondary: 11B30
paper · pdf · doi:10.48550/arxiv.2506.20459
openalex publication_date 2025/06/25 · openalex created_date 2025/10/09 · openalex updated_date 2026/07/28
The joint ergodicity classification problem aims to characterize those sequences which are jointly ergodic along an arbitrary dynamical system if and only if they satisfy two natural, simpler-to-verify conditions on this system. These two conditions, dubbed the difference and product ergodicity conditions, naturally arise from Berend and Bergelson's pioneering work on joint ergodicity. Elaborating on our earlier work, we investigate this problem for Hardy sequences of polynomial growth, this time without making any independence assumptions on the sequences. Our main result establishes the "difficult" direction of the problem: if a Hardy family satisfies the difference and product ergodicity conditions on a given system, then it is jointly ergodic for this system. We also find that, surprisingly, the converse fails for certain pathological families of Hardy sequences, even though it holds for all "reasonable" Hardy families. We conclude by suggesting potential fixes to the statement of this problem. New ideas of independent interest developed in this paper include the structure theory of a family of factors generalizing Host-Kra and box factors; a strengthening of Tao-Ziegler's concatenation results; and the most robust extension of a seminorm smoothing argument.