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Relations between chain recurrent points and turning points on the interval


Author: Shi Hai Li
Journal: Proc. Amer. Math. Soc. 115 (1992), 265-270
MSC: Primary 58F20; Secondary 26A18, 58F08
DOI: https://doi.org/10.1090/S0002-9939-1992-1079896-4
MathSciNet review: 1079896
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1079896-4
Keywords: Chain recurrent point, nonwandering point, recurrent point, turning point, $ \omega $-limit point, $ \alpha $-limit point, $ \gamma $-limit point
Article copyright: © Copyright 1992 American Mathematical Society

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