Relations Between Chain Recurrent Points and Turning Points on the Interval

Abstract

If a point is in the $\omega$-limit set and the $\alpha$-limit set of the same point, then we call it a $\gamma$-limit point. Then a $\gamma$-limit point is an $\omega$-limit point and thus a nonwandering point. In this paper, we prove that, on the interval, a nonwandering point which is not a $\gamma$-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 $\omega$-limit point forward image of some turning point. Two examples are given. One shows that a chain recurrent point which is not nonwandering, a $\gamma$-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 $\omega$-limit point which is not a $\gamma$-limit point can be a limit point of forward images of turning points but not a forward image nor an $\omega$-limit point of any turning point.