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)

$ω (x, f) \subseteq \bar{Per (f)}$ \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)

$Ω (f) = \bar{Per (f)}$ \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.

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On Limit Sets of Monotone Maps on Dendroids

E.N. Makhrova

Pubblicato online: 30 nov 2020

Pagine: 311 - 316

Ricevuto: 24 dic 2019

Accettato: 14 mar 2020

DOI: https://doi.org/10.2478/amns.2020.2.00056

Parole chiavedendroid, dendrite, monotone map, periodic point, non-wandering point, -limit set

© 2020 E.N. Makhrova, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Parole chiave
dendroid, dendrite, monotone map, periodic point, non-wandering point, -limit set