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
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.
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Berend, D. Multiple ergodic theorems. J. Anal. Math. 50, 123–142 (1988). https://doi.org/10.1007/BF02796117
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DOI: https://doi.org/10.1007/BF02796117