Chaotic functions with zero topological entropy.

*(English)*Zbl 0639.54029In “Period three implies chaos”, Am. Math. Monthly 82, 985-992 (1975; Zbl 0351.92021) T. Y. Li and J. A. Yorke introduced a notion of chaos for mappings of a compact interval to itself. Mappings with positive topological entropy exhibit his type of chaotic behaviour. The author is concerned with maps with zero topological entropy. S is called a \(\delta\)-scrambled set for f for some \(\delta\geq 0\) if for any x,y\(\in S\) \(x\neq y\), and any periodic point p we have
\[
\limsup_{n\to \infty}| f^ n(x)-f^ n(y)| >\delta,\quad \liminf_{n\to \infty}| f^ n(x)-f^ n(y)| =0\quad and\quad \limsup_{n\to \infty}| f^ n(x)-f^ n(p)| >\delta.
\]
f is chaotic if it has an infinite scrambled set. Any map with a cycle whose order is not a power of 2 is chaotic, while maps which do not have such a cycle either have cycles of finitely many different orders which are powers of 2 or have cycles of infinitely many such orders. The latter classes are said to be of type \(2^{\infty}\). The properties of such maps are studied. The main result is a characterization of when they are chaotic and the relationship with infinite attractors (the limit points of trajectories). Finally an example is given of a \(2^{\infty}\) map which is chaotic and an example is given of a \(2^{\infty}\) map which has an infinite attractor but which is non chaotic.

Reviewer: M.Sears