Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

$Z^d$ Toeplitz arrays

Pages: 859 - 881, Volume 15, Issue 3, July 2006      doi:10.3934/dcds.2006.15.859

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María Isabel Cortez - Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170/3 Correo 3, Santiago, Chile (email)

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.

Keywords:  Almost 1-1 extensions, Toeplitz, tiling, odometer.
Mathematics Subject Classification:  Primary: 54H20; Secondary: 37B50.

Received: January 2005;      Revised: November 2005;      Available Online: April 2006.