Equicontinuous local dendrite maps
Authors: Salem Aymen Haj , Hattab Hawete , Rejeiba Tarek 
Published: April 2021

<p>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:</p><p>(1) f is equicontinuous;</p><p>(2) <img src="data:image/png;base64,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" alt="" /> f<sup>n</sup> (X) = R(f);</p><p>(3) f| <img src="data:image/png;base64,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" alt="" /> f<sup>n</sup> (X) is equicontinuous;</p><p>(4) f| <img src="data:image/png;base64,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" alt="" />f<sup>n</sup> (X) is a pointwise periodic homeomorphism or is topologically conjugate to an irrational rotation of S 1 ;</p><p>(5) ω(x, f) = Ω(x, f) for all x ∈ X.</p><p>This result generalizes [17, Theorem 5.2], [24, Theorem 2] and [11, Theorem 2.8].</p>