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$Z^d$ Toeplitz arrays

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  • In this paper we give a definition of Toeplitz sequences and odometers for $\mathbb{Z}^d$ actions for $d\geq 1$ which generalizes that in dimension one. For these new concepts we study properties of the induced Toeplitz dynamical systems and odometers classical for $d=1$. In particular, we characterize the $\mathbb{Z}^d$-regularly recurrent systems as the minimal almost 1-1 extensions of odometers and the $\mathbb{Z}^d$-Toeplitz systems as the family of subshifts which are regularly recurrent.
    Mathematics Subject Classification: Primary: 54H20; Secondary: 37B50.


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