Equicontinuous local dendrite maps

Aymen Haj Salem, Hawete Hattab, Tarek Rejeiba

Abstract

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].


Keywords

dendrite; equicontinuity; local dendrite; recurrent point

Subject classification

37B05; 37B20; 37B45

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doi: 10.1007/s12346-021-00477-7



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