On Limit Sets of Monotone Maps on Dendroids

Let X be a dendrite, f : X → X be a monotone map. In the papers by I. Naghmouchi (2011, 2012) it is shown that ω -limit set ω ( x, f ) of any point x ∈ X has the next properties: (1) ω ( x , f ) ⊆ Per ( f ) ¯ \omega (x,f) \subseteq \overline {Per(f)} , where Per ( f ) is the set of periodic points of f ; (2) ω ( x, f ) is either a periodic orbit or a minimal Cantor set. In the paper by E. Makhrova, K. Vaniukova (2016 ) it is proved that (3) Ω ( f ) = Per ( f ) ¯ \Omega (f) = \overline {Per(f)} , where Ω( f ) is the set of non-wandering points of f . The aim of this note is to show that the above results (1) – (3) do not hold for monotone maps on dendroids.