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 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 $d_n$ (resp. ${\hat{d}}_n$) 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 ${\hat{d}}_n$ 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.

38 pages. Adding Appendix B (An Example by Cyr and Kra) and some references. To appear in Ergodic Theory Dynam. Systems

References in corpus (8)

• Mean equicontinuity and mean sensitivity
• Rate distortion theory, metric mean dimension and measure theoretic entropy
• Nonexpansive Z^2 subdynamics and Nivat's conjecture
• Measure-theoretic sensitivity via finite partitions
• Mean proximality and mean Li-Yorke chaos
• Measure complexity and Möbius disjointness
• Geometry and Dynamics of Admissible Metrics in Measure Spaces
• On the isomorphism problem for non-ergodic systems with discrete spectrum

Cited by in corpus (10)

• On the properties of the mean orbital pseudo-metric
• On Feldman-Katok metric and entropy formulae
• Measure-theoretic equicontinuity and rigidity
• Measure-theoretic sequence entropy pairs and mean sensitivity
• Mean equicontinuity and mean sensitivity on cellular automata
• Sequence entropy for amenable group actions
• A note on mean equicontinuity
• Local non-periodic order and diam-mean equicontinuity on cellular automata
• A note on logarithmic mean equicontinuity
• Discrete spectrum for amenable group actions