Description
| - Let $Q$ be the Cantor middle third set, and $S$ the circle, and let $\tau :Q\rightarrow Q$ be an adding machine (i.e., odometer). Let $X=Q\times S$ be equipped with (a metric equivalent to) the Euclidean metric. We show that there are continuous triangular maps $F_i: X\rightarrow X$, $F_i: (x,y)\mapsto (\tau (x), g_i(x,y))$, $i=1,2$, 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].
- Let $Q$ be the Cantor middle third set, and $S$ the circle, and let $\tau :Q\rightarrow Q$ be an adding machine (i.e., odometer). Let $X=Q\times S$ be equipped with (a metric equivalent to) the Euclidean metric. We show that there are continuous triangular maps $F_i: X\rightarrow X$, $F_i: (x,y)\mapsto (\tau (x), g_i(x,y))$, $i=1,2$, 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]. (en)
- Nechť $Q$ je Cantorova množina, $S$ kružnice a $\tau :Q\rightarrowQ$ je zobrazení adding machine. Na prostoru $X=Q\times S$ uvažujme Euklidovu metriku. Ukážeme, že existují zobrazení $F_i:X\rightarrow X$, $F_i: (x,y)\mapsto (\tau (x), g_i(x,y))$, $i=1,2$ 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]. (cs)
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