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

TL;DR

This paper investigates directional bounded complexity, mean equicontinuity, and discrete spectrum for $bZ^q$-actions, establishing equivalences between complexity notions, equicontinuity, and spectral properties in a multidimensional setting.

Contribution

It introduces new characterizations of bounded complexity and equicontinuity along directions for $bZ^q$-systems, linking these to spectral properties and extending classical concepts.

Findings

Bounded topological complexity is equivalent to directional equicontinuity.

Invariant measures with bounded complexity correspond to directional equicontinuity.

Bounded complexity with respect to certain metrics characterizes $bV$-discrete spectrum.

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.