Permutations, periodicity, and chaos

AbstractLet f: I → I be a continuous function on a closed interval I. If there exists x ϵ I which has period 3 with respect to f, then Li and Yorke [1] proved that f is chaotic in the sense that there are not only points x ϵ I of arbitrarily large period, but also uncountably many points x ϵ I which are not even asymptotically periodic with respect to f. By using only elementary combinatorial facts about permutations, it is shown that if there is a point x ϵ I of period p with respect to f, where p is divisible by 3, 5, or 7, then f is chaotic. The proof is followed by a study of some related combinatorial problems in symmetric groups