Finite odometer factors of rank one $\mathbb{Z}^d$-actions

Abstract

In this paper, we give explicit conditions characterizing the F{\o}lner rank one $\mathbb{Z}^d$-actions that factor onto a finite odometer; those that factor onto an arbitrary, but specified $\mathbb{Z}^d$-odometer, and those that factor onto an unspecified $\mathbb{Z}^d$-odometer. We also give explicit conditions describing the F{\o}lner rank one $\mathbb{Z}^d$-actions that are conjugate to a specific $\mathbb{Z}^d$-odometer, and those that are conjugate to some $\mathbb{Z}^d$-odometer. These conditions are based on cutting and stacking procedures used to generate the action, and generalize results given in \cite{FGHSW} for rank one $\mathbb{Z}$-actions.