Bounded complexity, mean equicontinuity and discrete spectrum

We study dynamical systems which have bounded complexity with respect to three kinds metrics: the Bowen metric $d_n$, the max-mean metric ${\widehat{d}}_n$ 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. ${\widehat{d}}_n$) 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 ${\widehat{d}}_n$ 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.