A note on chaotic maps

Abstract

It is well-known that, if a continuous map f of a closed interval into itself has a prime periodic point, then it is chaotic. The converse is not true. In this note, it is shown that a necessary and sufficient condition for f to have a prime periodic point is that it is chaotic and there is a chaotic set consisting of only nonwandering points. Let f be a continuous map of the closed interval I into itself. f is said to be chaotic (in Li-Yorke's sense) if 1. (1) f has infinitely many periods (of periodic points); 2. (2) there is an (uncountable) chaotic set S ⊂ I, i.e., an uncountable set S ⊂ I having the following properties: 3. (2)(a) for any x, y ϵ S, with x ≠ y, there exist sequences m i , n i , such that f m i ( x) and f m i ( y) converge to the same point but f n i ( x) and f n i ( y) converge to different points, 4. (2)(b) for every x ϵ S and every p ϵ the set p( f) of periodic point of f, there exists a sequence k i such that f k i ( x) and f k i ( p) converge to different points.