ﻻ يوجد ملخص باللغة العربية
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.
In this paper, directional sequence entropy and directional Kronecker algebra for $mathbb{Z}^q$-systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, direcitonal discrete spectrum s
In this paper we study the topological characteristic factors along cubes of minimal systems. It is shown that up to proximal extensions the pro-nilfactors are the topological characteristic factors along cubes of minimal systems. In particular, for
We study dynamical systems that have bounded complexity with respect to three kinds of directional metrics: the directional Bowen metric $d_k^{vec{v},b}$, the directional max-mean metric $hat{d}_k^{vec{v},b}$ and the directional mean metric $overline
We study directional mean dimension of $mathbb{Z}^k$-actions (where $k$ is a positive integer). On the one hand, we show that there is a $mathbb{Z}^2$-action whose directional mean dimension (considered as a $[0,+infty]$-valued function on the torus)
The counting and (upper) mass dimensions are notions of dimension for subsets of $mathbb{Z}^d$. We develop their basic properties and give a characterization of the counting dimension via coverings. In addition, we prove Marstrand-type results for bo