Equicontinuous local dendrite maps
DOI:
https://doi.org/10.4995/agt.2021.13446Keywords:
dendrite, equicontinuity, local dendrite, recurrent pointAbstract
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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