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

Published online by Cambridge University Press:  05 September 2012

TIM AUSTIN
TIM AUSTIN*
Affiliation:
Mathematics Department, Brown University, Box 1917, 151 Thayer Street, Providence, RI 02912, USA (email: [email protected])
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Abstract

Let $G$ 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 6 , December 2013 , pp. 1667 - 1708
Copyright
Copyright © 2012 Cambridge University Press 

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