Transitivity Implies Period Six: A Simple Proof

Abstract

In [4], Block and Coven showed that any continuous and transitive interval map must have a period-6 cycle. In this article, we give an elementary proof of the result and also show by example that six is the maximal period (in the Sharkovsky ordering) that transitivity can imply. Let I be a closed interval and let f : I -I be a map. A point p in I is called a period-n point if fn (p) = p and f'(p) :A p for 0 01, is called a period-n cycle. A continuous map f : I -I is called (topologically) transitive if there is a point whose orbit is dense in I. This property is equivalent to the fact that, for any given pair of nonempty open sets U and V in I, there is a positive integer n such that fn(U) n V :A 0. (This is called the Birkhoff Transitivity Theorem, the proof of which can be found in Theorem VII.2.1 of [8].) In [10], Vellekoop and Berglund showed that, for any continuous interval map, transitivity implies dense periodic points. In [2], Banks et al. proved that, for any continuous map on a metric space, transitivity and dense periodic points imply sensitive dependence on initial conditions. Therefore, every continuous and transitive interval map is chaotic in the sense of Devaney [5]; i.e., it exhibits transitivity plus dense periodic points plus sensitive dependence on initial conditions. In [9], Sharkovsky ordered all positive integers as follows: