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If a point is in the      ω  \omega      -limit set and the      α  \alpha      -limit set of the same point, then we call it a      γ  \gamma      -limit point. Then a      γ  \gamma      -limit point is an      ω  \omega      -limit point and thus a nonwandering point. In this paper, we prove that, on the interval, a nonwandering point which is not a      γ  \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      ω  \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      γ  \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      ω  \omega      -limit point which is not a      γ  \gamma      -limit point can be a limit point of forward images of turning points but not a forward image nor an      ω  \omega      -limit point of any turning point.

Relations between chain recurrent points and turning points on the interval