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.
Categories: math.DS