Recurrence of integral zeros for ergodic flows

TL;DR

This paper proves that for ergodic flows preserving a measure, points with non-zero function values return to a set with integral zero at an increasing sequence of times, with the integral of the function over these times also zero.

Contribution

It establishes the recurrence of integral zeros in ergodic flows for almost all points in a set with positive measure.

Findings

Almost all points with non-zero function values return to the set with zero integral at some sequence of times.

The integral of the function over these return times is exactly zero.

The result holds for flows preserving an ergodic probability measure.

Abstract

Let a flow $T_t$ preserve an ergodic probability measure $\mu$, $\int f\,d\mu=0$, and $\mu(A)>0$. Then for almost all $x\in A$, for which $f(x)\neq 0$, there is a sequence ${t_k}\to \infty$ such that $T_{t_k}x\in A$ and $\int_0^{t_k} f(T_sx)ds=0$.