Around Krygin-Atkinson, Shneiberg theorems, the recurrence with zero integrals

2024/11/20 by Valery V. Ryzhikov, Ryzhikov, Valery V.
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum chaos and dynamical systems #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2411.13486

openalex publication_date 2024/11/20 · openalex created_date 2024/11/24 · openalex updated_date 2026/07/28

Abstract

We recall theorems by Krygin, Atkinson, Shneiberg and propose the following assertion. Let T_t be an ergodic flow on (X,μ), let a function f on X have zero mean, and μ(A)>0 for A⊂ X. Then for almost all x∈ A with f(x)≠ 0 there exists a sequence t_k→∞ such that ∫₀^t_kf(T_sx)ds=0 and T_tkx∈ A.

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