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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, directional 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 consider weak horseshoe with bounded-gap-hitting times. For a flow $$(M,\phi )$$ , it is shown that if the time one map $$(M,\phi _1)$$ has weak horseshoe with bounded-gap-hitting times, so is $$(M,\phi _\tau )$$ for all $$\tau \ne 0$$ . In addition, we prove that for an affine homeomorphism of a compact metric abelian group, positive topological entropy is equivalent to weak horseshoe with bounded-gap-hitting times.

Let $\mathcal{M}(X)$ be the space of Borel probability measures on a compact metric space $X$ endowed with the weak$^\ast$-topology. In this paper, we prove that if the topological entropy of a nonautonomous dynamical system $(X,\{f_n\}_{n=1}^{+\infty})$ vanishes, then so does that of its induced system $(\mathcal{M}(X),\{f_n\}_{n=1}^{+\infty})$; moreover, once the topological entropy of $(X,\{f_n\}_{n=1}^{+\infty})$ is positive, that of its induced system $(\mathcal{M}(X),\{f_n\}_{n=1}^{+\infty})$ jumps to infinity. In contrast to Bowen's inequality, we construct a nonautonomous dynamical system whose topological entropy is not preserved under a finite-to-one extension., Comment: Accepted for publication in Journal of Differential Equations

By using a decomposition result for ergodic measure-preserving system with quasi-discrete spectrum, we prove that a generic point of an ergodic quasi-discrete spectrum measure in a topological dynamical system satisfies the required disjointness condition in Sarnak's Mobius Disjointness Conjecture. As a direct application, we have that Sarnak's Mobius Disjointness Conjecture holds for any topological model of an ergodic measure-preserving system with quasi-discrete spectrum.

In this paper, we consider measure-theoretical restricted sensitivity and topological restricted sensitivities by restricting the first sensitive time. For a given topological dynamical system, we define measure-theoretical restricted asymptotic rate with respect to sensitivity, and obtain that it equal to the reciprocal of the Brin-Katok local entropy for almost every point. For topological version we have similar definitions and conclusions.

In this paper we study ergodic measures of intermediate entropy for affine transformations of nilmanifolds. We prove that if an affine transformation \begin{document}$ \tau $\end{document} of nilmanifold has a periodic point, then for every \begin{document}$ a\in[0, h_{top}(\tau)] $\end{document} there exists an ergodic measure \begin{document}$ \mu_a $\end{document} of \begin{document}$ \tau $\end{document} such that \begin{document}$ h_{\mu_a}(\tau) = a $\end{document} .

We study dynamical systems which have bounded complexity with respect to three kinds metrics: the Bowen metric $d_n$, the max-mean metric $\hat{d}_n$ and the mean metric $\bar{d}_n$, both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to $d_n$ (resp. $\hat{d}_n$) if and only if it is equicontinuous (resp. equicontinuous in the mean). However, we construct minimal systems which have bounded complexity with respect to $\bar{d}_n$ but not equicontinuous in the mean. It turns out that an invariant measure $\mu$ on $(X,T)$ has bounded complexity with respect to $d_n$ if and only if $(X,T)$ is $\mu$-equicontinuous. Meanwhile, it is shown that $\mu$ has bounded complexity with respect to $\hat{d}_n$ if and only if $\mu$ has bounded complexity with respect to $\bar{d}_n$ if and only if $(X,T)$ is $\mu$-mean equicontinuous if and only if it has discrete spectrum., Comment: 38 pages. Adding Appendix B (An Example by Cyr and Kra) and some references. To appear in Ergodic Theory Dynam. Systems

We consider positive entropy G-systems for certain countable, discrete, infinite left-orderable amenable groups G. We will use local analysis to study the connection between the existence of asymptotic pairs, chaotic sets and stable sets. Examples are given for the case of integer lattice groups, the Heisenberg group, and the groups of integral unipotent upper triangular matrices.

The repetition property of a dynamical system, a notion introduced in Boshernitzan and Damanik (Commun Math Phys 283:647–662, 2008), plays an importance role in analyzing spectral properties of ergodic Schrodinger operators. In this paper, entropy of dynamical systems with repetition property is investigated. It is shown that the topological entropy of dynamical systems with the global repetition property is zero. Minimal dynamical systems having both topological repetition property and positive topological entropy are constructed. This provides a class of ergodic Schrodinger operators with potentials generated by positive entropy minimal dynamical systems that, in contrast to common beliefs, admit no eigenvalues.