## Abstract

The study of jointly ergodic measure preserving transformations of probability spaces, begun in [1], 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 σ_{1}σ_{2},...,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^{ s}converges 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.

Original language | English |
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Pages (from-to) | 255-284 |

Number of pages | 30 |

Journal | Journal d'Analyse Mathematique |

Volume | 45 |

Issue number | 1 |

DOIs | |

State | Published - 1 Dec 1985 |

Externally published | Yes |

## ASJC Scopus subject areas

- Analysis
- General Mathematics