The ergodic theorem with time compression
Classical ergodic theorems deal mainly with Cesàro-averages. Recently, averages taken with a nonlinear time scale or along subsequences have received considerable attention. E.g., Bourgain showed that if \(\tau\) is measure-preserving in a probability space, f bounded measurable and \(k_ j=j^ 2\), then \(f_ n=\frac{1}{n}\sum^{n}_{j=1}f\circ \tau^{k_ j}\) converges a.e. The present author presents a very novel counterpart of this result. He considers a measure-preserving flow \(\{\tau^ t,- \infty <t<\infty \}\) and shows that \(f_ n\) converges a.e. for \(k_ j=j^{\rho}\) with \(0<\rho <1\) even for all \(f\in L_ 1\). The result extends to more general concave functions of t of the kind used to define Orlicz spaces. There also is a maximal inequality.
scientific article; zbMATH DE number 4081886
scientific article