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Zbl 0894.60001
Lacey, Michael T.
The return time theorem fails on infinite measure-preserving systems. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 33, No.4, 491-495 (1997). ISSN 0246-0203

Summary: The return time theorem of {\it 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 $(X,{\cal A},\mu,T)$ 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 $(Y,{\cal B},\nu,S)$, 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 2000:: *60A10 Probabilistic measure theory
28D05 Measure-preserving transformations

Keywords: ergodic measure-preserving system; sigma-finite measure-preserving system; aperiodic measure-preserving system

Citations: Zbl 0705.28008

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