Aymen Haj Salem, Hawete Hattab, Tarek Rejeiba
Let X be a local dendrite, and f : X → X be a map. Denote by E(X) the set of endpoints of X. We show that if E(X) is countable, then the following are equivalent:(1) f is equicontinuous;(2) fn (X) = R(f);(3) f| fn (X) is equicontinuous;(4) f| fn (X) is a pointwise periodic homeomorphism or is topologically conjugate to an irrational rotation of S 1 ;(5) ω(x, f) = Ω(x, f) for all x ∈ X.This result generalizes [17, Theorem 5.2], [24, Theorem 2] and [11, Theorem 2.8].
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