Multiple ergodic theorems Academic Article uri icon

abstract

  • Given measure preserving transformationsT 1,T 2,...,T s of a probability space (X,B, μ) we are interested in the asymptotic behaviour of ergodic averages of the form \frac1Nån = 0N - 1 T1n f1 T2n f2 ¼Tsn fs \frac{1}{N}\sum\limits_{n = 0}^{N - 1} {T_1^n f_1 \cdot T_2^n f_2 } \cdot \cdots \cdot T_s^n f_s wheref 1,f 2,...,f s ɛL ∞(X,B,μ). In the general case we study, mainly for commuting transformations, conditions under which the limit of (1) inL 2-norm is ∫ x f 1 dμ·∫ x f 2 dμ...∫ x f s dμ for anyf 1,f 2...,f s ɛL ∞(X,B,μ). If the transformations are commuting epimorphisms of a compact abelian group, then this limit exists almost everywhere. A few results are also obtained for some classes of non-commuting epimorphisms of compact abelian groups, and for commuting epimorphisms of arbitrary compact groups.

publication date

  • January 1, 1988