Directional bounded complexity, mean equicontinuity and discrete spectrum for $\mathbb{Z}^q$-actions

Abstract: Given $q\in\mathbb{N}$, let $(X,T)$ be a $\mathbb{Z}^q$-system, $\vec{v}\in\mathbb{R}^q\setminus{\vec{0}}$ be a direction vector and $\textbf{b}\in\mathbb{R}+^{q-1}$. We study $(X,T)$ that has bounded complexity with respect to three kinds of metrics defined along direction $\vec{v}$: the directional Bowen metric $d_k^{\vec{v},\textbf{b}}$, the directional max-mean metric $\hat{d}_k^{\vec{v},\textbf{b}}$ and the directional mean metric $\bar{d}_k^{\vec{v},\textbf{b}}$. It is shown that $(X,T)$ has bounded topological complexity with respect to ${d_k^{\vec{v},\textbf{b}}}{k=1}^{\infty}$ (resp. ${\hat{d}k^{\vec{v},\textbf{b}}}{k=1}^{\infty}$) if and only if $T$ is $(\vec{v},\textbf{b})$-equicontinuous (resp. $(\vec{v},\textbf{b})$-equicontinuous in the mean). Meanwhile, it turns out that an invariant Borel probability measure $\mu$ on $X$ has bounded complexity with respect to ${d_k^{\vec{v},\textbf{b}}}_{k=1}^{\infty}$ if and only if $T$ is $(\mu,\vec{v},\textbf{b})$-equicontinuous. Moreover, it is shown that $\mu$ has bounded complexity with respect to ${\bar{d}k^{\vec{v},\textbf{b}}}{k=1}^{\infty}$ if and only if $\mu$ has bounded complexity with respect to ${\hat{d}k^{\vec{v},\textbf{b}}}{k=1}^{\infty}$ if and only if $T$ is $(\mu,\vec{v},\textbf{b})$-mean equicontinuous if and only if $T$ is $(\mu,\vec{v},\textbf{b})$-equicontinuous in the mean if and only if $\mu$ has $\vec{v}$-discrete spectrum.