On the existence of maximal ω-limit sets for dendrite maps

For interval maps and also for graph maps, every ω-limit set is a subset of a maximal one. In this note we construct a continuous map on a dendrite with no maximal ω-limit set. Moreover, the set of branch points is nowhere dense, every ω-limit set of the map is nowhere dense, the set of periodic points and the set of recurrent points are equal and the set of ω-limit points is not closed (an example with the last property was constructed by the authors already in [Kočan Z, Kornecká-Kurková V, Málek M. On the centre and the set of omega-limit points of continuous maps on dendrites. Topol Appl 2009;156:2923–2931]).