On dynamics of graph maps with zero topological entropy

Abstract

We explore the dynamics of graph maps with zero topological entropy. It is shown that a continuous map $f$ on a topological graph $G$ has zero topological entropy if and only if it is locally mean equicontinuous, that is the dynamics on each orbit closure is mean equicontinuous. As an application, we show that Sarnak's Möbius Disjointness Conjecture is true for graph maps with zero topological entropy. We also extend several results known in interval dynamics to graph maps. We show that a graph map has zero topological entropy if and only if there is no $3$-scrambled tuple if and only if the proximal relation is an equivalence relation; a graph map has no scrambled pairs if and only if it is null if and only if it is tame.

arXiv comment15 pages, updated version, add some references

References in corpus (7)

• Mean equicontinuity and mean sensitivity
• Recent development of chaos theory in topological dynamics
• Chaos and Entropy for Interval Maps
• Oscillating Sequences, Minimal Mean Attractability and Minimal Mean-Lyapunov-Stability
• Measure-theoretic sensitivity via finite partitions
• Möbius disjointness for topological models of ergodic systems with discrete spectrum
• Zero Entropy Interval Maps And MMLS-MMA Property

Cited by in corpus (3)

• Mobius disjointness conjecture for local dendrite maps
• Quasi-graphs, zero entropy and measures with discrete spectrum
• Some properties of circle maps with zero topological entropy