Article
On Limit Sets of Monotone Maps on Dendroids
Authors
Abstract
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.
Keywords
dendroid, dendrite, monotone map, periodic point, non-wandering point, ω-limit set, 54H20, 54F50, 37B20, 37C25
Citation
Makhrova, E. (2020). On limit sets of monotone maps on dendroids. Applied Mathematics and Nonlinear Sciences, 5(2), 311–316. https://doi.org/10.2478/amns.2020.2.00056
E. Makhrova, “On limit sets of monotone maps on dendroids,” Applied Mathematics and Nonlinear Sciences, vol. 5, no. 2, pp. 311–316, 2020, doi: 10.2478/amns.2020.2.00056.
Makhrova E. On limit sets of monotone maps on dendroids. Applied Mathematics and Nonlinear Sciences. 2020;5(2):311–316. doi:10.2478/amns.2020.2.00056.
Makhrova, E. (2020), ‘On limit sets of monotone maps on dendroids’, Applied Mathematics and Nonlinear Sciences, 5(2), pp. 311–316. Available at: https://doi.org/10.2478/amns.2020.2.00056.
Makhrova, E.n. “On Limit Sets of Monotone Maps on Dendroids.” Applied Mathematics and Nonlinear Sciences, vol. 5, no. 2, 2020, pp. 311–316. https://doi.org/10.2478/amns.2020.2.00056.
Makhrova, E.n. “On Limit Sets of Monotone Maps on Dendroids.” Applied Mathematics and Nonlinear Sciences 5, no. 2 (2020): 311–316. https://doi.org/10.2478/amns.2020.2.00056.
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Published by: Engineering Journals
