Counterexamples to double recurrence for non-commuting deterministic transformations

2025/07/21 by Zemer Kosloff, Kosloff, Zemer, Shrey Sanadhya +1
Computer Science · Mathematics · #28D05 #37A05 #37A30 #37A50 #60F05 #60G10 #60G15 #Advanced Combinatorial Mathematics #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2507.15528

openalex publication_date 2025/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that if <i>p</i>₁, <i>p</i>₂ are injective, integer polynomials that vanish at the origin, such that either both are of degree 1 or both are of degree 2 or higher, then double recurrence fails for non-commuting, mixing, zero entropy transformations. This answers a question of Frantzikinakis and Host completely.

Citations

• Spectra and joint dynamics of Poisson suspensions for rank-one automorphisms
• Non-commuting transformations with non-converging 2-fold ergodic averages
• A counterexample on multiple convergence without commutativity
• Non-convergence of some non-commuting double ergodic averages
• Polynomial Fourier decay and a cocycle version of Dolgopyat's method for self conformal measures
• A counterexample on polynomial multiple convergence without commutativity
• Multiple recurrence and convergence without commutativity
• Local limit theorem in deterministic systems

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