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On dynamics of graph maps with zero topological entropy. (English) Zbl 1380.37088

Summary: 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.

MSC:

37E25 Dynamical systems involving maps of trees and graphs
37B40 Topological entropy
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
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