The study of jointly ergodic measure preserving transformations of probability spaces, begun in , is continued, and notions of joint weak and strong mixing are introduced. Various properties of ergodic and mixing transformations are shown to admit analogues for several transformations. The case of endomorphisms of compact abelian groups is particularly emphasized. The main result is that, given such commuting endomorphisms σ1σ2,...,σ, of G, the sequence ((1/N)Σ n=0 N-1 σ 1 n f 1·σ 2 n f 2· ··· · σ s n f sconverges in L 2(G) for every f 1, f 2,..., f s∈L ∞(G). If, moreover, the endomorphisms are jointly ergodic, i.e., if the limit of any sequence as above is Π i=1 s ∫ G f 1 d μ, where μ is the Haar measure, then the convergence holds also μ-a.e.
ASJC Scopus subject areas
- Mathematics (all)