Bounded complexity, mean equicontinuity and discrete spectrum

We study dynamical systems that 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 $d_{n}$ (respectively $\hat{d}_{n}$ ) if and only if it is equicontinuous (respectively equicontinuous in the mean). However, we construct minimal systems that have bounded complexity with respect to $\bar{d}_{n}$ but that are not equicontinuous in the mean. It turns out that an invariant measure $\unicode[STIX]{x1D707}$ on $(X,T)$ has bounded complexity with respect to $d_{n}$ if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$ -equicontinuous. Meanwhile, it is shown that $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\hat{d}_{n}$ if and only if $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\bar{d}_{n}$ , if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$ -mean equicontinuous and if and only if it has discrete spectrum.