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Abstract <p> Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(T_t\)</EquationSource> </InlineEquation> be a measure-preserving measurable ergodic flow on a probability space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((X,\mu)\)</EquationSource> </InlineEquation>. Suppose given a zero-mean function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(f\colon X\to\mathbb R\)</EquationSource> </InlineEquation> and a set <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(A\subset X\)</EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mu(A)&gt;0\)</EquationSource> </InlineEquation>. Then, for almost all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(x\in A\)</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(f(x)\neq 0\)</EquationSource> </InlineEquation>, there exists a sequence <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(t_k\to\infty\)</EquationSource> </InlineEquation> satisfying the conditions <Equation ID="Equi"> <EquationSource Format="TEX">\(\int_0^{t_k} f(T_s x)\,ds=0, \qquad T_{t_k}x\in A.\)</EquationSource> </Equation></p>

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Recurrence of Integral Zeros for Ergodic Flows

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • V. V. Ryzhikov

摘要

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 \(\int_0^{t_k} f(T_s x)\,ds=0, \qquad T_{t_k}x\in A.\)