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The return time theorem fails on infinite measure-preserving systems. (English) Zbl 0894.60001
Summary: The return time theorem of J. Bourgain [Publ. Math., Inst. Hautes Étud. Sci. 69, 5-45 (1989; Zbl 0705.28008)] cannot be extended to the infinite measure-preserving case. Specifically, there exist a sigma-finite measure-preserving system \((X,{\mathcal A},\mu,T)\) and a set \(A\subset X\) of positive finite measure so that for almost every \(x\in X\) the following undesirable behavior occurs. For every aperiodic measure-preserving system \((Y,{\mathcal B},\nu,S)\), with \(\nu(S)=1\), there is a square-integrable \(g\) on \(Y\) so that the averages \(\tau_n^{-1} \sum_{m=1}^n 1_A(T^mx) g(S^my)\) diverge a.e. \((y)\), where \(\tau_n= \tau_n(x)= \sum_{m=1}^n 1_A(T^n x)\).

MSC:
60A10 Probabilistic measure theory
28D05 Measure-preserving transformations
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