Three forms of chaos and their associated attractors
There are different ways how to define chaos if we consider a continuous map of a real compact interval into itself. Namely, a map can be regarded as chaotic if its topological entropy is positive or if it possesses a scrambled set in the sense of Li and Yorke. A further possibility can be obtained by measuring the separation of trajectories via the map \(\omega_ f: x\to\omega(x,f)\), where \(\omega(x,f)\) is the set of all cluster points of the trajectory \(\{f^ n(x)\}^ \infty_{n=0}\). The property \(\omega_ f\not\in B_ 1\), where \(B_ 1\) is the first Baire class, can serve as a third definition of chaos which is strictly intermediate to the first two mentioned. The author investigates the existence of certain kinds of \(\omega\)-limit sets for functions chaotic in all discussed senses. Among others, a necessary and sufficient condition for \(\omega_ f\not\in B_ 1\) is given.
scientific article; zbMATH DE number 54526