On Devaney Chaos and Dense Periodic Points: Period 3 and Higher Implies Chaos

Abstract

We look at density of periodic points and Devaney Chaos. We prove that if f is Devaney Chaotic on a compact metric space with no isolated points, then the set of points with prime period at least n is dense for each n. Conversely, we show that if f is a continuous function from a closed interval to itself, for which the set of points with prime period at least n is dense for each n, then there is a decomposition of the interval into closed subintervals on which either f or f2 is Devaney Chaotic. (In fact, this result holds if the set of points with prime period at least 3 is dense.)