an:02012361
Zbl 1045.37004
Akin, Ethan; Kolyada, Sergi??
Li-Yorke sensitivity
EN
Nonlinearity 16, No. 4, 1421-1433 (2003).
00095291
2003
j
37B05 54H20
Li-Yorke chaos; proximal points; minimal system
The Li-Yorke definition of chaos is linked here to the natural notion of sensitivity to initial conditions. A topological dynamical system \((X,T)\) is said to be Li-Yorke sensitive if there exists \(\varepsilon>0\) with the property that every point \(x\in X\) is a limit of points \(y\) for which \((x,y)\) is proximal but not \(\varepsilon\)-asymptotic. Li-Yorke sensitivity is strictly stronger than sensitivity: a minimal system which is distal but not equicontinuous is sensitive but not Li-Yorke sensitive. Here, it is shown that a topologically weak-mixing system is Li-Yorke sensitive (it was known earlier that such systems are Li-Yorke chaotic). In addition a system is constructed which is Li-Yorke chaotic but not Li-Yorke sensitive. Several open problems are raised about the structure of Li-Yorke sensitive maps.
Thomas Ward (Norwich)