Bounded complexity, mean equicontinuity and 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 metricd n and the mean metric 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.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 tod n but not equicontinuous in the mean.It turns out that an invariant measure µ on (X, T ) has bounded complexity with respect to d n if and only if (X, T ) is µ-equicontinuous. Meanwhile, it is shown that µ has bounded complexity with respect tod n if and only if µ has bounded complexity with respect tod n if and only if (X, T ) is µ-mean equicontinuous if and only if it has discrete spectrum.

Measure-theoretic mean equicontinuity and bounded complexity

“…In this section, we show that for an invariant measure µ on (X , T ), (X , T ) is µ-mean equicontinuous if and only if the complexity of (X , B X , µ, T ) using α-names of a partition and the Hamming distance is bounded. It should be noticed that Huang et al [14,Theorem 4.2,4.3,4.6] proved that (X , T ) is µ-mean equicontinuous if and only if it has discrete spectrum.…”

“…Thus M is a mean equicontinuous set and (X , T ) is µ-mean equicontinuous with the metric ρ. By [14,Theorem 4.2,4.3,4.6], we know that (X , T ) is µ-mean equicontinuous if and only if (X , T ) has discrete spectrum, which does not rely on the metric. So (X , T ) is µ-mean equicontinuous with metric d.…”

“…(X , T ) and showed that (X , T, µ) is µ-mean equicontinuous if and only if it has discrete spectrum, see also [20] for another proof. It should be noticed that recently the authors in [14,17] showed that for an invariant measure µ on a t.d.s. (X , T ), (X , T, µ) is µ-mean equicontinuous if and only if it has discrete spectrum.…”

“…In [3], Blanchard et al studied topological complexity via the complexity function of an open cover and showed that the complexity function is bounded for any open cover if and only if the system is equicontinuous. In [14,17], Huang et al studied topological and measure-theoretic complexity via a sequence of metrics induced by a metric and showed that an invariant measure µ on (X , T ) has bounded complexity with respect tod n if and only if (X , T ) is µ-mean equicontinuous if and only if it has discrete spectrum, whered n (x, y) = 1 n ∑ n−1 i=0 d (T i x, T i y). Ferenczi [5] studied measure-theoretic complexity of a m.p.s (X , B, µ, T ) using αnames of a partition and the Hamming distance.…”

Mean equicontinuity, complexity and applications

“…This result was generalized to countable discrete abelian group action at the end of [15], and to continuous abelian locally compact group action in [18]. We note that Huang et al [28] proved that this result still holds without the ergodicity assumption.…”