Equidistribution of joinings under off-diagonal polynomial flows of nilpotent Lie groups (Q2874001)

This paper provides a general equidistribution result for polynomial actions of connected nilpotent Lie groups on joinings of its actions. Results of this sort are motivated by \textit{H. Furstenberg}'s ergodic multiple recurrence theorem [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)], which may be phrased in terms of joinings. Let \(\lambda\) be a joining of \(k+1\) actions of a connected nilpotent Lie group \(G\); in particular \(\lambda\) is invariant under the diagonal action of \(G\) on the Cartesian product of the \(k+1\) spaces, but is not necessarily invariant under the jointly measurable action of the whole Cartesian product \(G^{k+1}\). The main result of the paper is that, for any one-parameter subgroup of \(G^{k+1}\), the trajectory of the joining \(\lambda\) under the action of the subgroup equidistributes with respect to some new joining \(\lambda'\) that is also invariant under the subgroup. More generally, the same equidistribution holds for averages over any map \(\mathbb R\to G^{k+1}\) that is `polynomial' (that is, has the property that repeated differencing using the group structure leads to the trivial map in finitely many steps).