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Register a new userDirectional Kronecker algebra for $mathbb{Z}^q$-actions
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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.
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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.
In this paper, entropies, including measure-theoretic entropy and topological entropy, are considered for random $mathbb{Z}^k$-actions which are generated by random compositions of the generators of $mathbb{Z}^k$-actions. Applying Pesins theory for commutative diffeomorphisms we obtain a measure-theoretic entropy formula of $C^{2}$ random $mathbb{Z}^k$-actions via the Lyapunov spectra of the generators. Some formulas and bounds of topological entropy for certain random $mathbb{Z}^k$(or $mathbb{Z}_+^k$ )-actions generated by more general maps, such as Lipschitz maps, continuous maps on finite graphs and $C^{1}$ expanding maps, are also obtained. Moreover, as an application, we give a formula of Friedlands entropy for certain $C^{2}$ $mathbb{Z}^k$-actions.
We obtain a sufficient condition for a substitution ${mathbb Z}$-action to have pure singular spectrum in terms of the top Lyapunov exponent of the spectral cocycle introduced in arXiv:1802.04783 by the authors. It is applied to a family of examples, including those associated with self-similar interval exchange transformations.
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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