Equidistribution of joinings under off-diagonal polynomial flows of nilpotent Lie groups

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.