An explicit description of all scrambled sets of weakly unimodal functions of type \(2^ \infty \)
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.
scientific article; zbMATH DE number 1095859
scientific article