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Joint ergodicity for group actions

Published online by Cambridge University Press:  19 September 2008

Vitaly Bergelson  and
Joseph Rosenblatt
Vitaly Bergelson
Affiliation:
The Ohio State University, Columbus, Ohio 43210, USA
Joseph Rosenblatt
Affiliation:
The Ohio State University, Columbus, Ohio 43210, USA
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Abstract

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Let T1,…,Tn be continuous representations of a σ-compact separable locally compact amenable group G as measure-preserving transformations of a non-atomic separable probability space (X, β, m). Let (Kn) be a right Følner sequence of compact sets in G. If T1,…,Tn are pairwise commuting in the sense that Ti(g)Tj(h) = Tj(h)Ti(g) for i ≠ j and g, h ∈ G, then necessary and sufficient conditions can be given, in terms of the ergodicity of certain tensor products, for the following to hold: for all F1,…,Fn∈L∞, the sequence AN(x) where

converges in L2(X) to . The necessary and sufficient conditions are that each of the following representations are ergodic: Tn, Tn−1⊗Tn−1Tn,…,T2⊗T2T3⊗…⊗T2…Tn, T1⊗T1T2⊗…⊗T1…Tn.

In order to prove this theorem, specific properties of the decomposition of L2(X) into its weakly mixing and compact subspaces with respect to a representation Ti are needed. These properties are also used to prove some generalizations of wellknown facts from ergodic theory in the case where G is the integer group Z.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 3 , September 1988 , pp. 351 - 364
Copyright
Copyright © Cambridge University Press 1988

References

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