On convergence of the averages \({1\over N}\sum_{n=1}^N f_1(R^nx)f_2(S^nx)f_3(T^nx)\)
Given three commuting ergodic measure-preserving transformations \(R,S,T\) satisfying the additional property that \(RT^{-1},ST^{-1},TR^{-1}\) are ergodic, this paper proves the following non-conventional ergodic theorem: \({1\over N}\sum_{n=1}^{N} f_1(R^nx)f_2(S^nx)f_3(T^nx)\) converges in \(L^1\). The main ingredients of the proof include an analysis of the `Conze-Lesigne' algebra [\textit{J.-P. Conze} and \textit{E. Lesigne}, Bull. Soc. Math. Fr. 112, 143-175 (1984; Zbl 0595.28018)] and the circle of ideas developed by \textit{H. Furstenberg} [J. Anal. Math. 31, 204-256 (1977; Zbl 0347.28016)]. The results strengthens that of \textit{H. Furstenberg} and \textit{Y. Katznelson} [J. Anal. Math. 34, 275-291 (1978; Zbl 0426.28014)].
commuting measure-preserving transformation
scientific article; zbMATH DE number 1051738
scientific article