Equicontinuous local dendrite maps

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