About: A zero topological entropy map with recurrent points not $F\sb \sigma$

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
Description	We show that there is a continuous map $\chi$ of the unit interval into itself of type $2^\infty$ which has a trajectory disjoint from the set $ \operatorname{Rec}(\chi )$ of recurrent points of $\chi$, but contained in the closure of $ \operatorname{Rec}(\chi )$. In particular, $ \operatorname{Rec}(\chi )$ is not closed. A function $\psi$ of type $2^\infty$, with nonclosed set of recurrent points, was found by H. Chu and J. Xiong [Proc. Amer. Math. Soc. 97 (1986), 361-366]. However, there is no trajectory contained in $\overline {\operatorname{Rec} (\psi)}\setminus \operatorname{Rec}(\psi)$, since any point in $\overline { \operatorname{Rec}(\psi)}$ is eventually mapped into $\operatorname{Rec} (\psi)$. Moreover, our construction is simpler. We use $\chi$ to show that there is a continuous map of the interval of type $2^\infty$ for which the set of recurrent points is not an $F_\sigma$ set. This example disproves a conjecture of A. N. Sharkovsky-1989 We show that there is a continuous map $\chi$ of the unit interval into itself of type $2^\infty$ which has a trajectory disjoint from the set $ \operatorname{Rec}(\chi )$ of recurrent points of $\chi$, but contained in the closure of $ \operatorname{Rec}(\chi )$. In particular, $ \operatorname{Rec}(\chi )$ is not closed. A function $\psi$ of type $2^\infty$, with nonclosed set of recurrent points, was found by H. Chu and J. Xiong [Proc. Amer. Math. Soc. 97 (1986), 361-366]. However, there is no trajectory contained in $\overline {\operatorname{Rec} (\psi)}\setminus \operatorname{Rec}(\psi)$, since any point in $\overline { \operatorname{Rec}(\psi)}$ is eventually mapped into $\operatorname{Rec} (\psi)$. Moreover, our construction is simpler. We use $\chi$ to show that there is a continuous map of the interval of type $2^\infty$ for which the set of recurrent points is not an $F_\sigma$ set. This example disproves a conjecture of A. N. Sharkovsky-1989 (en)
Title	A zero topological entropy map with recurrent points not $F\sb \sigma$ A zero topological entropy map with recurrent points not $F\sb \sigma$ (en)
skos:prefLabel	A zero topological entropy map with recurrent points not $F\sb \sigma$ A zero topological entropy map with recurrent points not $F\sb \sigma$ (en)
skos:notation	RIV/47813059:19610/03:00000121!RIV/2004/MSM/196104/N
http://linked.open.../vavai/riv/strany	2089;209
http://linked.open...avai/riv/aktivita	P Z
http://linked.open...avai/riv/aktivity	P(GA201/00/0859), Z(MSM 192400002)
http://linked.open...iv/cisloPeriodika	7
http://linked.open...vai/riv/dodaniDat	2004
http://linked.open...aciTvurceVysledku	Augustová, Petra
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	Slezská univerzita v Opavě / Matematický ústav v Opavě
http://linked.open...dnocenehoVysledku	597228
http://linked.open...ai/riv/idVysledku	RIV/47813059:19610/03:00000121
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	Topological entropy; recurrent points; periodic points; $\omega$-limit s (en)
http://linked.open.../riv/klicoveSlovo	periodic points $\omega$-limit s Topological entropy recurrent points
http://linked.open...odStatuVydavatele	US - Spojené státy americké
http://linked.open...ontrolniKodProRIV	[75909DF05355]
http://linked.open...i/riv/nazevZdroje	Proceedings of the American Mathematical Society
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	1 (xsd:int)
http://linked.open...ocetUcastnikuAkce	0 (xsd:int)
http://linked.open...nichUcastnikuAkce	0 (xsd:int)
http://linked.open...vavai/riv/projekt	Dynamical systems
http://linked.open...UplatneniVysledku	2003
http://linked.open...v/svazekPeriodika	131
http://linked.open...iv/tvurceVysledku	Šindelářová, Petra
http://linked.open...n/vavai/riv/zamer	http://linked.opendata.cz/resource/domain/vavai/zamer/MSM%20192400002
issn	0002-9939
number of pages	8 (xsd:int)
http://localhost/t...ganizacniJednotka	19610

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