Maps of the interval with closed periodic set

Abstract

We 8how that for any continuou8 map of the interval who8e periodic point8 form a clo8ed 8et, every nonwandering point i8 periodic with least period a power of two. Block showed [Bi] that if the set Per(f) of periodic points for a continuous map of the interval f: I -I is finite and consists only of fixed points, then the nonwandering set Ql(f) equals Per(f). Coven and Hedlund [CH] extended this, obtaining the same conclusion from the weaker hypothesis that some power g = fnl of f simultaneously fixes all the periodic points. Other results related to these are established in [B2, CH, L]. In this paper, we extend the results stated above. THEOREM. If f: I -I is continuous and Per(f) is a closed set, then Q(f) Per(f). I would like to thank Ethan Coven and Louis Block for useful conversations about this problem, including a gap in my original version. I have heard recently of an independent proof of the theorem above by Jin-Cheng Xiong [X]. We note that Per(f) closed does not imply that the set of least periods is finite, so that Coven-Hedlund's result need not apply. To construct an example, we simply string together maps fn: [, 1 ] -+ [ -n ] so that fn(k) = k and Per(f,) contains points of least period 2n, but none higher. We will see from the proof of our theorem that this example is in essence the only situation in which Per(f) is closed and [CH] does not apply. Our point of departure is Block's homoclinic point theorem [B3]. Given a periodic point p for f and a power of f, g = fn, define the full, left, and right unstable sets of the g-orbit of p by w(p, g) n u gk (p ,p +, E>O k>O (1) WW(p,g,L) nf U gk(p_,p], E>O k>O WU(p,g,R)= n U gk[p,pc) E>O k>O It is clear that when g(p) = p, each unstable set is an interval containing p (and perhaps nothing else). One can see that WU(p, g) = WU(p, g, L) U WU(p, g, R). Received by the editors August 25, 1981. 1980 Mathematics Subject Classification. Primary 58F20, 54H20.