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Register a new userDirectional dynamical cubes for minimal $mathbb{Z}^{d}$-systems
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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.
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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 systems and directional null systems are defined. It is shown that a $mathbb{Z}^q$-system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a $mathbb{Z}^q$-system has directional discrete spectrum along $q$ linearly independent directions if and only if it has discrete spectrum.
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 a distal minimal system, the maximal $(d-1)$-step pro-nilfactor is the topological cubic characteristic factor of order $d$.
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 $overline{d}_k^{vec{v},b}$. It is shown that a $mathbb{Z}^q$-topological system $(X,T)$ has bounded topological complexity with respect to ${d_k^{vec{v},b}}$ (respectively ${hat{d}_k^{vec{v},b}}$) if and only if $T$ is $(vec{v},b)$-equicontinuous (respectively $(vec{v},b)$-equicontinuous in the mean). It turns out that a measure $mu$ has bounded complexity with respect to ${d_k^{vec{v},b}}$ if and only if $T$ is $(mu,vec{v},b)$-equicontinuous. Whats more, it is shown that $mu$ has bounded complexity with respect to ${overline{d}_k^{vec{v},b}}$ if and only if $mu$ has bounded complexity with respect to ${hat{d}_k^{vec{v},b}}$ if and only if $T$ is $(mu,vec{v},b)$-mean equicontinuous if and only if $T$ is $(mu,vec{v},b)$-equicontinuous in the mean if and only if $mu$ has $vec{v}$-discrete spectrum.
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) is not continuous. On the other hand, we prove that if a $mathbb{Z}^k$-action is continuum-wise expansive, then the values of its $(k-1)$-dimensional directional mean dimension are bounded. This is a generalization (with a view towards Meyerovitch and Tsukamotos theorem on mean dimension and expansive multiparameter actions) of a classical result due to Ma~ne: Any compact metrizable space admitting an expansive homeomorphism (with respect to a compatible metric) is finite-dimensional.
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 both dimensions. For example, if $A subseteq mathbb{R}^d$ has counting dimension $D(A)$, then for almost every orthogonal projection with range of dimension $k$, the counting dimension of the image of $A$ is at least $min big(k,D(A)big)$. As an application, for subsets $A_1, ldots, A_d$ of $mathbb{R}$, we are able to give bounds on the counting and mass dimensions of the sumset $c_1 A_1 + cdots + c_d A_d$ for Lebesgue-almost every $c in mathbb{R}^d$. This work extends recent work of Y. Lima and C. G. Moreira.
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