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 ;

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.