For dynamical systems generated by continuous maps of a compact interval, the centre of the dynamical system is a subset of the set of ω-limit points which is closed. This holds even for graphs. In this paper we provide an example of a continuous self-map f1 of a dendrite such that ω(f1) is not closed, and it 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. Moreover, we construct a triangular map of the square with
the same property. Again, this is impossible for continuous maps on a compact interval.