On the centre and the set of ω-limit points of continuous maps on dendrites

It is well known that for dynamical systems generated by continuous maps of a graph, the centre of the dynamical system is a subset of the set of ω-limit points.In this paper we provide an example of a continuous self-map f1 of a dendrite such that ω(f1) is a proper subset of C(f1).The second example is a continuous self-map f2 of a dendrite having a strictly increasing sequence of ω-limit sets which is not contained in any maximal one. Again, this is impossible for continuous maps on graphs.