Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages

2024/09/08 by Zemer Kosloff, Kosloff, Zemer, Shrey Sanadhya +1
Mathematics · #28D05 #37A05 #37A30 #37A50 #60F05 #60G10 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Equations Stability Results #Nonlinear Differential Equations Analysis #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2409.05087

openalex publication_date 2024/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that for every ergodic and aperiodic probability preserving system (X,B,m,T), there exists f:X→ ℤ^d, whose corresponding cocycle satisfies the d-dimensional local central limit theorem. We use the 2-dimensional result to resolve a question of Huang, Shao and Ye and Franzikinakis and Host regarding non-convergence in L² of polynomial multiple averages of non-commuting zero entropy transformations. Our methods also give the first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates.

Related

• CF-Nil systems and convergence of two-dimensional ergodic averages
• Pointwise convergence for cubic and polynomial ergodic averages of non-commuting transformations
• Local limit theorem in deterministic systems
• On the pointwise convergence of multiple ergodic averages
• A counterexample on polynomial multiple convergence without commutativity