Bounded complexity, mean equicontinuity and discrete spectrum

TL;DR

This paper investigates bounded complexity in dynamical systems using three metrics, establishing links to equicontinuity, mean equicontinuity, and discrete spectrum, with new examples and characterizations in topological and measure-theoretic contexts.

Contribution

It characterizes bounded complexity with respect to different metrics and relates these to equicontinuity, mean equicontinuity, and discrete spectrum, including constructing minimal systems with specific properties.

Findings

Bounded complexity w.r.t. $d_n$ iff system is equicontinuous.

Bounded complexity w.r.t. $ar{d}_n$ does not imply mean equicontinuity.

Invariant measure $oxed{ ext{has bounded complexity iff $oxed{ ext{system is $oxed{ ext{μ}- ext{equicontinuous}}$}}}$ and relates to discrete spectrum.

Abstract

We study dynamical systems which have bounded complexity with respect to three kinds metrics: the Bowen metric $d_n$, the max-mean metric $\hat{d}_n$ and the mean metric $\bar{d}_n$, both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to $d_n$ (resp. $\hat{d}_n$) if and only if it is equicontinuous (resp. equicontinuous in the mean). However, we construct minimal systems which have bounded complexity with respect to $\bar{d}_n$ but not equicontinuous in the mean. It turns out that an invariant measure $\mu$ on $(X,T)$ has bounded complexity with respect to $d_n$ if and only if $(X,T)$ is $\mu$-equicontinuous. Meanwhile, it is shown that $\mu$ has bounded complexity with respect to $\hat{d}_n$ if and only if $\mu$ has bounded complexity with respect to $\bar{d}_n$ if and only if $(X,T)$ is $\mu$-mean equicontinuous if and only if it has discrete spectrum.