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) boolean AND(infinity)(n=1) f(n) (X) =R(f);
(3) f vertical bar boolean AND(infinity)(n=1) f(n)(X) is equicontinuous;
(4)f vertical bar boolean AND(infinity)(n=1) f(n) (X) is a pointwise periodic homeomorphism or is topologically conjugate to an irrational rotation of S-1;
(5) omega(x, f) = Omega(x, f) for all x is an element of X.
This result generalizes [17, Theorem 5.2], [24, Theorem 2] and [11, Theorem 2.8].