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.
A. For a ∈ [0, +∞), the function space E ≥a (E > a ; E ≤a ; E < a ) of all continuous maps from [0, 1] to itself whose topological entropies are larger than or equal to a (larger than a; smaller than or equal to a; smaller than a) with the supremum metric is investigated. It is shown that the spaces E ≥a and E > a are homeomorphic to the Hilbert space l 2 and the spaces E ≤a and E < a are contractible. Moreover, the subspaces of E ≤a and E < a consisting of all piecewise monotone maps are homotopy dense in them, respectively.2010 Mathematics Subject Classification. 37E05, 54F65, 54H20.
<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M2">\begin{document}$ (X,T) $\end{document}</tex-math></inline-formula> be a topological dynamical system and <inline-formula><tex-math id="M3">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>. We say that <inline-formula><tex-math id="M4">\begin{document}$ (X,T) $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive (resp. <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive) if there exists a constant <inline-formula><tex-math id="M7">\begin{document}$ \delta>0 $\end{document}</tex-math></inline-formula> with the property that for each non-empty open subset <inline-formula><tex-math id="M8">\begin{document}$ U $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M9">\begin{document}$ X $\end{document}</tex-math></inline-formula>, there exist <inline-formula><tex-math id="M10">\begin{document}$ x_1,x_2,\dotsc,x_n\in U $\end{document}</tex-math></inline-formula> such that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Bigl\{k\in \mathbb{N}\colon \min\limits_{1\le i<j\le n}d(T^k x_i,T^k x_j)>\delta\Bigr\} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is an IP-set (resp. a thick set).</p><p style='text-indent:20px;'>We obtain several sufficient and necessary conditions of a dynamical system to be <inline-formula><tex-math id="M11">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive or <inline-formula><tex-math id="M12">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive and show that any non-trivial weakly mixing system is <inline-formula><tex-math id="M13">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive for all <inline-formula><tex-math id="M14">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>, while it is <inline-formula><tex-math id="M15">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive if and only if it has at least <inline-formula><tex-math id="M16">\begin{document}$ n $\end{document}</tex-math></inline-formula> minimal points. We characterize two kinds of sensitivity by considering some kind of factor maps. We introduce the opposite side of pairwise IP-sensitivity and pairwise thick sensitivity, named (almost) pairwise IP<inline-formula><tex-math id="M17">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuity and (almost) pairwise syndetic equicontinuity, and obtain dichotomies results for them. In particular, we show that a minimal system is distal if and only if it is pairwise IP<inline-formula><tex-math id="M18">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuous. We show that every minimal system admits a maximal almost pairwise IP<inline-formula><tex-math id="M19">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuous factor and admits a maximal pairwise syndetic equicontinuous factor, and characterize them by the factor maps to their maximal distal factors.</p>
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