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Published online by Cambridge University Press: 02 April 2001
We define staircase $\mathbb{Z}^d$ actions. We first prove that staircase $\mathbb{Z}^2$ actions satisfying a general condition are mixing. Then we describe how to extend the results to the staircase $\mathbb{Z}^d$ actions. Thus we have constructed explicitly rank one mixing $\mathbb{Z}^d$ actions which include natural analogues to the well-known staircase transformation.