Multiple ergodic theorems

Given measure preserving transformations T 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 {Mathematical expression} where f ₁, f ₂,..., f _s e{open}L ^∞(X, B,μ). In the general case we study, mainly for commuting transformations, conditions under which the limit of (1) in L ²-norm is ∫ _x f ₁ dμ·∫ _x f ₂ dμ...∫ _x f _s dμ for any f ₁, f ₂..., f _s e{open}L ^∞(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.