An explicit description of all scrambled sets of weakly unimodal functions of type \(2^ \infty \). (English) Zbl 0879.58044

Let \(f\) be a weakly unimodal map of type \(2^\infty\) on the interval \(I\). For any \(x,y\in I\) write \(x\sim^sy\) if \(\lim_{n\rightarrow \infty} |f^n(x)-f^n(y)|=0\), and \(x\sim^iy\) if \(\liminf_{n\rightarrow \infty} |f^n(x)-f^n(y)|=0\). Clearly, chaos in the sense of Li and Yorke can be expressed in terms of both relations: there are \(x,y\in I\) such that \(x\sim^iy\), but not \(x\sim^sy\). The main contribution is the study of relations \(\sim^s\) and \(\sim^i\) on the set \(K(f)\) of points with infinite \(\omega\)-limit sets. Using a special kind of coding, the author gives an alternative characterization of chaotic maps of the above mentioned type, and of their maximal scrambled sets. The paper contains some open problems.
Reviewer: J.Smítal (Opava)


37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
26A18 Iteration of real functions in one variable
54H20 Topological dynamics (MSC2010)


chaos; scrambled set