On dynamics of graph maps with zero topological entropy

TL;DR

This paper characterizes zero topological entropy in graph maps through local mean equicontinuity, extends interval dynamics results, and confirms Sarnak's conjecture for these systems.

Contribution

It provides a new characterization of zero entropy graph maps via local mean equicontinuity and extends key results from interval dynamics to graph maps.

Findings

Zero entropy iff the system is locally mean equicontinuous

Sarnak's M"obius Disjointness Conjecture holds for zero entropy graph maps

Equivalence between absence of 3-scrambled tuples, nullness, and tameness

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 M\"obius 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.