Directional dynamical cubes for minimal -systems
We introduce the notions of directional dynamical cubes and directional regionally proximal relation defined via these cubes for a minimal $\mathbb{Z}^{d}$-system $(X,T_{1},\ldots ,T_{d})$. We study the structural properties of systems that satisfy the so-called unique closing parallelepiped property and we characterize them in several ways. In the distal case, we build the maximal factor of a $\mathbb{Z}^{d}$-system $(X,T_{1},\ldots ,T_{d})$ that satisfies this property by taking the quotient with respect to the directional regionally proximal relation. Finally, we completely describe distal $\mathbb{Z}^{d}$-systems that enjoy the unique closing parallelepiped property and provide explicit examples.
1988 ◽
Vol 447
(3)
◽
pp. 103-116
◽
1984 ◽
Vol 127
(1-3)
◽
pp. 214-218
◽
Keyword(s):
Keyword(s):
1998 ◽
Vol 08
(PR2)
◽
pp. Pr2-47-Pr2-50
Keyword(s):
2000 ◽
Vol 10
(PR7)
◽
pp. Pr7-95-Pr7-98
◽
2013 ◽
Vol 51
(9)
◽
pp. 691-699
Keyword(s):
1962 ◽
Vol 78
(12)
◽
pp. 579-617
◽
2007 ◽
Vol 2007
(suppl_26)
◽
pp. 503-508