A zero topological entropy map with recurrent points not $Fsb sigma$

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F47813059%3A19610%2F03%3A00000121" target="_blank" >RIV/47813059:19610/03:00000121 - isvavai.cz</a>

Výsledek na webu

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DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

A zero topological entropy map with recurrent points not $Fsb sigma$

Popis výsledku v původním jazyce

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

Název v anglickém jazyce

A zero topological entropy map with recurrent points not $Fsb sigma$

Popis výsledku anglicky

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

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

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Projekt

<a href="/cs/project/GA201%2F00%2F0859" target="_blank" >GA201/00/0859: Dynamické systémy</a><br>

Návaznosti

Z - Vyzkumny zamer (s odkazem do CEZ)

Rok uplatnění

2003

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Proceedings of the American Mathematical Society

ISSN

ISSN0002-9939

e-ISSN

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Svazek periodika

131

Číslo periodika v rámci svazku

7

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

8

Strana od-do

2089-209

Kód UT WoS článku

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EID výsledku v databázi Scopus

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