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

Abstract

AbstractLet$G$be a connected nilpotent Lie group. Given probability-preserving$G$-actions$(X_i,\Sigma _i,\mu _i,u_i)$,$i=0,1,\ldots ,k$, and also polynomial maps$\phi _i:\mathbb {R}\to G$,$i=1,\ldots ,k$, we consider the trajectory of a joining$\lambda $of the systems$(X_i,\Sigma _i,\mu _i,u_i)$under the ‘off-diagonal’ flow\[ (t,(x_0,x_1,x_2,\ldots ,x_k))\mapsto (x_0,u_1^{\phi _1(t)}x_1,u_2^{\phi _2(t)}x_2,\ldots ,u_k^{\phi _k(t)}x_k). \]It is proved that any joining$\lambda $is equidistributed under this flow with respect to some limit joining$\lambda '$. This is deduced from the stronger fact of norm convergence for a system of multiple ergodic averages, related to those arising in Furstenberg’s approach to the study of multiple recurrence. It is also shown that the limit joining$\lambda '$is invariant under the subgroup of$G^{k+1}$generated by the image of the off-diagonal flow, in addition to the diagonal subgroup.