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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 complexitywith respect to three kinds of metrics defined along direction $\vec{v}$: thedirectional Bowen metric $d_k^{\vec{v},\textbf{b}}$, the directional max-meanmetric $\hat{d}_k^{\vec{v},\textbf{b}}$ and the directional mean metric$\bar{d}_k^{\vec{v},\textbf{b}}$. It is shown that $(X,T)$ has boundedtopological 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 outthat an invariant Borel probability measure $\mu$ on $X$ has bounded complexitywith 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 $\mu$ hasbounded 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.

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