Li-Yorke sensitive minimal maps
Result description
Let $Q$ be the Cantor middle third set, and $S$ the circle, and let $tau :Qrightarrow Q$ be an adding machine (i.e., odometer). Let $X=Qtimes S$ be equipped with (a metric equivalent to) the Euclidean metric. We show that there are continuous triangular maps $F_i: Xrightarrow X$, $F_i: (x,y)mapsto (tau (x), g_i(x,y))$, $i=1,2$, with the following properties: Both $(X, F_1)$ and $(X, F_2)$ are minimal systems, without weak mixing factors (i.e., neither of them is semiconjugate to a weak mixing system). $(X, F_1)$ is spatio-temporally chaotic but not Li-Yorke sensitive. $(X,F_2)$ is Li-Yorke sensitive. This disproves conjectures of E. Akin and S. Kolyada [Li-Yorke sensitivity, {it Nonlinearity} 16 (2003), 1421--1433].
Keywords
Li-Yorke sensitiveminimal settriangular mapweak mixing systemspatio-temporally chaotic
The result's identifiers
Result code in IS VaVaI
Result on the web
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DOI - Digital Object Identifier
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Alternative languages
Result language
angličtina
Original language name
Li-Yorke sensitive minimal maps
Original language description
Let $Q$ be the Cantor middle third set, and $S$ the circle, and let $tau :Qrightarrow Q$ be an adding machine (i.e., odometer). Let $X=Qtimes S$ be equipped with (a metric equivalent to) the Euclidean metric. We show that there are continuous triangular maps $F_i: Xrightarrow X$, $F_i: (x,y)mapsto (tau (x), g_i(x,y))$, $i=1,2$, with the following properties: Both $(X, F_1)$ and $(X, F_2)$ are minimal systems, without weak mixing factors (i.e., neither of them is semiconjugate to a weak mixing system). $(X, F_1)$ is spatio-temporally chaotic but not Li-Yorke sensitive. $(X,F_2)$ is Li-Yorke sensitive. This disproves conjectures of E. Akin and S. Kolyada [Li-Yorke sensitivity, {it Nonlinearity} 16 (2003), 1421--1433].
Czech name
Li-Yorkova senzitivita minimálních funkcí
Czech description
Nechť $Q$ je Cantorova množina, $S$ kružnice a $tau :QrightarrowQ$ je zobrazení adding machine. Na prostoru $X=Qtimes S$ uvažujme Euklidovu metriku. Ukážeme, že existují zobrazení $F_i:Xrightarrow X$, $F_i: (x,y)mapsto (tau (x), g_i(x,y))$, $i=1,2$s následujícími vlastnostmi: Oba systémy $(X, F_1)$ i $(X, F_2)$ jsou minimální bez slabě mixujícího faktoru (tzn. neexistuje semikonjugace do slabě mixujícího systému). $(X, F_1)$ je spatio-temporally chaotický, ale není Li-Yorkovsky senzitivní. $(X, F_2)$ je Li-Yorkovský senzitivní. Toto vyvrací hypotézy z článku od E. Akina a S. Kolyady [Li-Yorke sensitivity, {it Nonlinearity} 16 (2003), 1421--1433].
Classification
Type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Result continuities
Project
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Z - Vyzkumny zamer (s odkazem do CEZ)
Others
Publication year
2006
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Nonlinearity
ISSN
0951-7715
e-ISSN
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Volume of the periodical
19
Issue of the periodical within the volume
2
Country of publishing house
GB - UNITED KINGDOM
Number of pages
13
Pages from-to
517-529
UT code for WoS article
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EID of the result in the Scopus database
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Result type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP
BA - General mathematics
Year of implementation
2006