In this paper, relationships between equicontinuity of a dendrite map f on Λ(f), absence of Li-Yorke pairs, collection of minimal sets and regularly recurrent points are investigated.
Let X be a dendrite with set of endpoints E(X) closed and let f : X → X be a continuous map with zero topological entropy. Let P (f ) be the set of periodic points of f . We prove that if L is an infinite ω-limit set of f then L ∩ P (f ) ⊂ E(X) , where E(X) is the set of all accumulations points of E(X). Furthermore, if E(X) is countable and L is uncountable then L ∩ P (f ) = ∅. We also show that if E(X) is finite then any uncountable ω-
We prove that the Möbius disjointness conjecture holds for graph maps with zero topological entropy and for all monotone local dendrite maps. We further show that this also holds for continuous maps on certain class of dendrites. Moreover, we see that there is an example of transitive dendrite map with zero entropy for which Möbius disjointness conjecture holds.
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