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Abstract
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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