### Abstract

Let G be a connected nilpotent Lie group. Given probability-preserving G-actions (Xi ; Σ_{i} ; μ_{i} ; μ_{i}), i = 0, 1;..., k, and also polynomial maps 'i V ℝ → G, i = 1, ... , k, we consider the trajectory of a joining λ of the systems (Xi ; Σ_{i} ; μ_{i} ; μ_{i}) under the 'off-diagonal' flow (eqution presented) It is proved that any joining is equidistributed under this flow with respect to some limit joining λ0. This is deduced from the stronger fact of norm convergence for a system of multiple ergodic averages, related to those arising in Furstenberg's approach to the study of multiple recurrence. It is also shown that the limit joining λ0 is invariant under the subgroup of GkC1 generated by the image of the off-diagonal flow, in addition to the diagonal subgroup.

Original language | English (US) |
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Pages (from-to) | 1667-1708 |

Number of pages | 42 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 33 |

Issue number | 6 |

DOIs | |

State | Published - Dec 2013 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Ergodic Theory and Dynamical Systems*,

*33*(6), 1667-1708. https://doi.org/10.1017/etds.2012.113