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

References in corpus (2)

• Mean equicontinuity and mean sensitivity
• Mean proximality and mean Li-Yorke chaos