Multiple ergodic theorems
Let s be a positive integer and \(T_ 1,...,T_ s\) be measure-preserving transformations of a probability space \((X,{\mathcal B},\mu).\) The author studies the asymptotic behaviour of ergodic averages \[ A_ n(f_ 1,...,f_ s)=\frac{1}{n}\sum^{n-1}_{k=0}T\quad k_ 1f_ 1\cdot...\cdot T\quad k_ sf_ s, \] where \(f_ 1,...,f_ s\in L^{\infty}\). The paper is in a sense a continuation of the articles by the author and \textit{V. Bergelson} [Isr. J. Math. 49, 307-314 (1984; Zbl 0571.28012)] and the author [J. Anal. Math. 45, 255-284 (1985; Zbl 0595.28020)]. Some results give, mainly for transformations commuting in pairs, conditions equivalent to joint ergodicity, i.e. to the relation \[ A_ n(f_ 1,...,f_ s)\to_{n\to \infty}\int_{X}f_ 1d\mu \cdot...\cdot \int_{X}f_ sd\mu \quad in\quad L\quad 2-norm,\quad f_ 1,...,f_ s\in L^{\infty}. \] Moreover, the author proves that, for commuting epimorphisms of compact Abelian groups, the sequence of the averages \(A_ n(f_ 1,...,f_ s)\) converges almost everywhere. The same is true in the case of arbitrary compact groups provided the given epimorphisms are assumed to be jointly ergodic. Also some special systems of not necessarily commuting group endomorphisms are studied.
asymptotic behaviour of ergodic averages
commuting epimorphisms of compact Abelian groups
scientific article; zbMATH DE number 4065352