Multiple recurrence and convergence without commutativity

2021/11/02 by Nikos Frantzikinakis, Frantzikinakis, Nikos, Bernard Host +1 · 2 citations
Materials Science · Mathematics · #28D05 #Dynamical Systems (math.DS) #FOS: Mathematics #Graph theory and applications #Mathematical Dynamics and Fractals #Primary: 37A30 #Quasicrystal Structures and Properties #Secondary: 37A45

paper · pdf · doi:10.48550/arxiv.2111.01518

openalex publication_date 2021/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish multiple recurrence and convergence results for pairs of zero entropy measure preserving transformations that do not satisfy any commutativity assumptions. Our results cover the case where the iterates of the two transformations are n and n^k respectively, where k≥ 2, and the case k=1 remains an open problem. Our starting point is based on the observation that Furstenberg systems of sequences of the form (f(T^nkx)) have very special structural properties when k≥ 2. We use these properties and some disjointness arguments in order to get characteristic factors with nilpotent structure for the corresponding ergodic averages, and then finish the proof using some equidistribution results on nilmanifolds.

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• CF-Nil systems and convergence of two-dimensional ergodic averages
• Counterexamples to double recurrence for non-commuting deterministic transformations

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