Li-Yorke sensitivity. (English) Zbl 1045.37004

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.


37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
54H20 Topological dynamics (MSC2010)
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