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 (d) over cap (n) and the mean metric (d) over bar (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 (d) over cap (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 (d) over bar (n) but that are not equicontinuous in the mean. It turns out that an invariant measure mu on (X, T) has bounded complexity with respect to dn if and only if (X, T) is mu-equicontinuous. Meanwhile, it is shown that mu has bounded complexity with respect to (d) over cap (n) if and only if mu has bounded complexity with respect to (d) over bar (n), if and only if (X; T / is mu-mean equicontinuous and if and only if it has discrete spectrum.