O některých vlastnostech zobrazení intervalu s nulovou topologickou entropií

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F47813059%3A19610%2F08%3A%230000219" target="_blank" >RIV/47813059:19610/08:#0000219 - 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

On some properties of interval maps with zero topological entropy

Popis výsledku v původním jazyce

We consider the following six properties of continuous maps of the compact interval: (i) $f$ has zero topological entropy; (ii) $Rec (f)$ is an $F_sigma$ set; (iii) $f$ is Lyapunov stable on $Per(f)$; (iv) for any $varepsilon > 0$, any infinite $omega$-limit set of $f$ has a cover consisting of disjoint compact periodic intervals with length less than $varepsilon$; (v) $Per (f)$ is a $G_delta$ set; (vi) every linearly ordered chain of $omega$-limit sets is countable. Some of these properties were basically studied in the sixties by A. N. Sharkovsky, and they were believed to be equivalent. But recently several authors have provided counterexamples. In this paper we complete these results, solve some open problems and disprove a recent conjecture. Thus, we show that (iv) $Rightarrow$ (iii) $Rightarrow$ (ii) $Rightarrow$ (i), (iv) $Rightarrow$ (vi) $Rightarrow$ (i), and (v) $Rightarrow$ (i), and that there is no other implication between these properties.

Název v anglickém jazyce

On some properties of interval maps with zero topological entropy

Popis výsledku anglicky

We consider the following six properties of continuous maps of the compact interval: (i) $f$ has zero topological entropy; (ii) $Rec (f)$ is an $F_sigma$ set; (iii) $f$ is Lyapunov stable on $Per(f)$; (iv) for any $varepsilon > 0$, any infinite $omega$-limit set of $f$ has a cover consisting of disjoint compact periodic intervals with length less than $varepsilon$; (v) $Per (f)$ is a $G_delta$ set; (vi) every linearly ordered chain of $omega$-limit sets is countable. Some of these properties were basically studied in the sixties by A. N. Sharkovsky, and they were believed to be equivalent. But recently several authors have provided counterexamples. In this paper we complete these results, solve some open problems and disprove a recent conjecture. Thus, we show that (iv) $Rightarrow$ (iii) $Rightarrow$ (ii) $Rightarrow$ (i), (iv) $Rightarrow$ (vi) $Rightarrow$ (i), and (v) $Rightarrow$ (i), and that there is no other implication between these properties.

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%2F06%2F0318" target="_blank" >GA201/06/0318: Dynamické systémy III</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>Z - Vyzkumny zamer (s odkazem do CEZ)

Rok uplatnění

2008

Kód důvěrnosti údajů

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

Název periodika

Aequationes Mathematicae

ISSN

0001-9054

e-ISSN

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Svazek periodika

76

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

10

Strana od-do

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Kód UT WoS článku

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EID výsledku v databázi Scopus

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