Relations between Chain Recurrent Points and Turning Points on the Interval

If a point is in the ¿u-limit set and the a-limit set of the same point, then we call it a y-limit point. Then a y-limit point is an w-limit point and thus a nonwandering point. In this paper, we prove that, on the interval, a nonwandering point which is not a y-limit point is in the closure of the set of forward images of turning points, and such points are not always the forward images of turning points. But a nonwandering point which is not an ¿u-limit point forward image of some turning point. Two examples are given. One shows that a chain recurrent point which is not nonwandering, a y-limit point which is not recurrent and a recurrent point which is not periodic need not be in the closure of forward images of turning points. The other shows that an ¿i)-limit point which is not a y-limit point can be a limit point of forward images of turning points but not a forward image nor an w-limit point of any turning point. Let / = [0, 1] be a compact interval on the real line. Let /:'/—►/ be a continuous map. For each x G /, we let Orb(x) = {y G /: y = /'(x), / > 0} . Let T = {x £ I: f is not a local homeomorphism at x} denote the set of turning points of / and Orb(T) = {y : y = fix), / > 0, x £ T} denote the set of forward images of turning points of /. A point x is called a periodic point if fix) = x for some n > 0. A point x G / is called an «y-limit (alimit) point if there is a y G / such that x is an accumulation point of {/"(y)} (there is a sequence of integers {«,} with n,• —» oo and a sequence of points {y„.} with /"'(y„,) = y such that x = lim,_00y„/). Denote the set of «-limit points (a-limit points) of x by cd(x) (a(x)). If x G w(y) Da(y) for some y, then x is called a y-limit point. A point x is called a chain recurrent point if for any number e > 0 there exists a sequence of points, Xo = x, xx, ... , xn-X , xn = x, such that |/(x;) xi+J | < e for / = 0, ... , n 1. CRif), Q(/), A(/), T(/), Rif), Pif) represent the collection of the chain recurrent points, nonwandering points, co-limit points, y-limit points, recurrent points, periodic points, respectively. Without causing confusion, we write CR, Í2, A, T, R, P simply. Since Xiong has proved that A2(/) = T(/) for any continuous map on the interval in [XI], where A2(/) = A(/|A), we will abuse A2(/) and T(/) Received by the editors September 30, 1990 and, in revised form, November 2, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 58F20; Secondary 26A18.