\(\mathbb Z^d\) Toeplitz arrays
In the paper under review the definitions of odometer and Toeplitz flow are extended to \(\mathbb Z^d\) actions \((d\geq 1)\); the special cases \(d=1,2\) were considered before in [\textit{K. Jacobs} and \textit{M. Keane}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 13, 123-131 (1969; Zbl 0195.52703)] and [\textit{T. Downarowicz}, Isr. J. Math. 97, 239-251 (1997; Zbl 0893.54032)], respectively. The main result is the characterization of \(\mathbb Z^d\)-regularly 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. Moreover two interesting \(\mathbb Z^2\)-Toeplitz flows are constructed: the first one with a finite number of ergodic mesures and the second one being uniquely ergodic with positive entropy.
scientific article; zbMATH DE number 5140554