Monotone preimages of dendrites

Abstract

A counterexample to a theorem of J.J. Charatonik and K. Omiljanowski giving sufficient conditions for a dendrite to be contained in all of its monotone preimages is given, and a corrected version of the theorem is presented. An alternative proof is provided for a characterization of dendrites monotonely equivalent to a universal dendrite that was originally proved using the erroneous result. Finally, in response to a question by J.J. Charatonik, it is shown that a dendrite contained in all of its monotone preimages must have a discrete set of ramification points.