About: On the $\omega$-limit sets of product maps

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
Description	Let $\omega(\cdot)$ denote the union of all $\omega$-limit sets of a given map. As the main result of this paper we prove that, for given continuous interval maps $f_1,\ldots, f_m$, the set of $\omega$-limit points of the product map $f_1 \times \cdots \times f_m$ and the cartesian product of the sets $\omega(f_1),\ldots, \omega(f_m)$ coincide. This result substantially enriches the theory of multidimensional permutation product maps, i.e., maps of the form $F(x_1,\ldots, x_m) = (f_{\sigma(1)}(x_{\sigma(1)}), \ldots,f_{\sigma(m)}(x_{\sigma(m)}))$, where $\sigma$ is a permutation of the set of indices $\{1,\ldots,m\}$. Especially, for any such map $F$, we prove that the set $\omega(F)$ is closed and we also show that $\omega(F)$ cannot be a proper subset of the center of the map $F$. These results solve open questions mentioned, e.g., in [F. Balibrea, J. S. C\'{a}novas, A. Linero, {\em New results on topological dynamics of antitriangular maps\/}, Appl. Gen. Topol.]. Let $\omega(\cdot)$ denote the union of all $\omega$-limit sets of a given map. As the main result of this paper we prove that, for given continuous interval maps $f_1,\ldots, f_m$, the set of $\omega$-limit points of the product map $f_1 \times \cdots \times f_m$ and the cartesian product of the sets $\omega(f_1),\ldots, \omega(f_m)$ coincide. This result substantially enriches the theory of multidimensional permutation product maps, i.e., maps of the form $F(x_1,\ldots, x_m) = (f_{\sigma(1)}(x_{\sigma(1)}), \ldots,f_{\sigma(m)}(x_{\sigma(m)}))$, where $\sigma$ is a permutation of the set of indices $\{1,\ldots,m\}$. Especially, for any such map $F$, we prove that the set $\omega(F)$ is closed and we also show that $\omega(F)$ cannot be a proper subset of the center of the map $F$. These results solve open questions mentioned, e.g., in [F. Balibrea, J. S. C\'{a}novas, A. Linero, {\em New results on topological dynamics of antitriangular maps\/}, Appl. Gen. Topol.]. (en)
Title	On the $\omega$-limit sets of product maps On the $\omega$-limit sets of product maps (en)
skos:prefLabel	On the $\omega$-limit sets of product maps On the $\omega$-limit sets of product maps (en)
skos:notation	RIV/61988987:17610/10:A1100RIW!RIV11-MSM-17610___
http://linked.open...avai/riv/aktivita	P Z
http://linked.open...avai/riv/aktivity	P(1M0572), Z(MSM4781305904)
http://linked.open...iv/cisloPeriodika	3-4
http://linked.open...vai/riv/dodaniDat	2011
http://linked.open...aciTvurceVysledku	Kupka, Jiří
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Ostravská univerzita v Ostravě / Centrum excelence IT4Innovations, divize OU, Ústav pro výzkum a aplikace fuzzy modelování
http://linked.open...dnocenehoVysledku	276894
http://linked.open...ai/riv/idVysledku	RIV/61988987:17610/10:A1100RIW
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	Discrete dynamical system; interval map; product map; permutation product map; antitriangular map; $\omega$-limit set; solenoidal set; basic set; center (en)
http://linked.open.../riv/klicoveSlovo	center interval map $\omega$-limit set Discrete dynamical system antitriangular map basic set permutation product map product map solenoidal set
http://linked.open...odStatuVydavatele	US - Spojené státy americké
http://linked.open...ontrolniKodProRIV	[5911BFE0B0F9]
http://linked.open...i/riv/nazevZdroje	Dynamic Systems and Applications
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	3 (xsd:int)
http://linked.open...vavai/riv/projekt	Data, algorithms, decision making
http://linked.open...UplatneniVysledku	2010
http://linked.open...v/svazekPeriodika	19
http://linked.open...iv/tvurceVysledku	Kupka, Jiří Jiménez López, Victor Linero, Antonio
http://linked.open...ain/vavai/riv/wos	000285265100019
http://linked.open...n/vavai/riv/zamer	Topological and analytical methods in the theory of dynamical systems and mathematical physics
issn	1056-2176
number of pages	12 (xsd:int)
http://localhost/t...ganizacniJednotka	17610
is http://linked.open...avai/riv/vysledek of	On the $\omega$-limit sets of product maps On the $\omega$-limit sets of product maps On the $\omega$-limit sets of product maps

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