Abstract:
Let $T_t$ be a measure-preserving measurable ergodic flow on a probability space $(X,\mu)$. Suppose given a zero-mean function $f\colon X\to\mathbb R$ and a set $A\subset X$ with $\mu(A)>0$. Then, for almost all $x\in A$ such that $f(x)\neq 0$, there exists a sequence $t_k\to\infty$ satisfying the conditions $$ \int_0^{t_k} f(T_s x)\,ds=0, \qquad T_{t_k}x\in A. $$
Keywords:recurrence, integrals along trajectories, cylindrical flow, ergodicity.
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