Bounded complexity, mean equicontinuity and discrete spectrum

We study dynamical systems that have bounded complexity with respect to three kinds metrics: the Bowen metric dₙ , the max-mean metric d̂ₙ and the mean metric d̄ₙ , both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to dₙ (respectively d̂ₙ ) if and only if it is equicontinuous (respectively equicontinuous in the mean). However, we construct minimal systems that have bounded complexity with respect to d̄ₙ but that are not equicontinuous in the mean. It turns out that an invariant measure [STIX]x1D707 on $(X,T)$ has bounded complexity with respect to dₙ if and only if $(X,T)$ is [STIX]x1D707 -equicontinuous. Meanwhile, it is shown that [STIX]x1D707 has bounded complexity with respect to d̂ₙ if and only if [STIX]x1D707 has bounded complexity with respect to d̄ₙ , if and only if $(X,T)$ is [STIX]x1D707 -mean equicontinuous and if and only if it has discrete spectrum.