Ergodic Theory and Dynamical Systems



Pointwise convergence of ergodic averages for polynomial actions of $\mathbb{Z}^{d}$ by translations on a nilmanifold


A. LEIBMAN a1
a1 Department of Mathematics, The Ohio State University, OH 43221, USA (e-mail: leibman@math.ohio-state.edu)

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Abstract

Generalizing the one-parameter case, we prove that the orbit of a point on a compact nilmanifold X under a polynomial action of $\mathbb{Z}^{d}$ by translations on X is uniformly distributed on the union of several sub-nilmanifolds of X. As a corollary we obtain the pointwise ergodic theorem for polynomial actions of $\mathbb{Z}^{d}$ by translations on a nilmanifold.

(Received February 5 2003)
(Revised February 3 2004)