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

Result code in 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>
Result on the web
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DOI - Digital Object Identifier
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Result language
angličtina
Original language name
A zero topological entropy map with recurrent points not $Fsb sigma$
Original language 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
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
<a href="/en/project/GA201%2F00%2F0859" target="_blank" >GA201/00/0859: Dynamical systems</a><br>
Continuities
Z - Vyzkumny zamer (s odkazem do CEZ)

Publication year
2003
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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