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

TL;DR

This paper discusses classical theorems in ergodic theory and introduces a new assertion about the recurrence of ergodic flows with zero mean functions, extending known results.

Contribution

It proposes a new recurrence result for ergodic flows with zero mean functions, building upon and extending theorems by Krygin, Atkinson, and Shneiberg.

Findings

Almost all points in a positive measure set return with zero integral for the flow.

Existence of a sequence where the integral of the function along the flow is zero.

Recurrence properties are established for functions with zero mean in ergodic systems.

Abstract

We recall theorems by Krygin, Atkinson, Shneiberg and propose the following assertion. Let $T_t$ be an ergodic flow on $(X,\mu)$, let a function $f$ on $X$ have zero mean, and $\mu(A)>0$ for $A\subset X$. Then for almost all $x\in A$ with $f(x)\neq 0$ there exists a sequence $t_k\to\infty$ such that $\int_0^{t_k}f(T_sx)ds=0$ and $T_{t_k}x\in A$.