<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>X</italic> be a dendrite, <italic>f</italic> : <italic>X</italic> → <italic>X</italic> be a monotone map. In the papers by I. Naghmouchi (2011, 2012) it is shown that <italic>ω</italic>-limit set <italic>ω</italic>(<italic>x, f</italic> ) of any point <italic>x</italic> ∈ <italic>X</italic> has the next properties:
<list list-type="custom"><list-item><label>(1)</label><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amns.2020.2.00056_ieq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⊆</mo><mover accent="true"><mrow><mi mathvariant="italic">Per</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><mo stretchy="true">¯</mo></mover></mrow></math><tex-math>\omega (x,f) \subseteq \overline {Per(f)} </tex-math></alternatives></inline-formula>
, where <italic>Per</italic>( <italic>f</italic> ) is the set of periodic points of <italic>f</italic> ;</list-item><list-item><label>(2)</label><italic>ω</italic>(<italic>x, f</italic> ) is either a periodic orbit or a minimal Cantor set.</list-item></list>In the paper by E. Makhrova, K. Vaniukova (2016 ) it is proved that
<list list-type="custom"><list-item><label>(3)</label><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amns.2020.2.00056_ieq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo>Ω</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mover accent="true"><mrow><mi mathvariant="italic">Per</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><mo stretchy="true">¯</mo></mover></mrow></math><tex-math>\Omega (f) = \overline {Per(f)} </tex-math></alternatives></inline-formula>
, where Ω( <italic>f</italic> ) is the set of non-wandering points of <italic>f</italic>.</list-item></list>The aim of this note is to show that the above results <xref ref-type="disp-formula" rid="j_amns.2020.2.00056_eq_001_w2aab3b7e2754b1b6b1ab1b2b3b5Aa">(1)</xref> – <xref ref-type="disp-formula" rid="j_amns.2020.2.00056_eq_003_w2aab3b7e2754b1b6b1ab1b2b5c32Aa">(3)</xref> do not hold for monotone maps on dendroids.</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.

On Limit Sets of Monotone Maps on Dendroids

Lobachevsky State University of Nizhny Novgorod

Applied Mathematics and Nonlinear Sciences

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

{"article-title":"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...

On Limit Sets of Monotone Maps on Dendroids

Published Online: Nov 30, 2020

Page range: 311 - 316

Received: Dec 24, 2019

Accepted: Mar 14, 2020

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

Keywords
dendroid, 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.

Fig. 1

On Limit Sets of Monotone Maps on Dendroids

Published Online: Nov 30, 2020

Page range: 311 - 316

Received: Dec 24, 2019

Accepted: Mar 14, 2020

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

Keywordsdendroid, 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.

Fig. 1

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