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

MathSciNet review:
1079896

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

**[B]**Louis Block,*Continuous maps of the interval with finite nonwandering set*, Trans. Amer. Math. Soc.**240**(1978), 221–230. MR**0474240**, 10.1090/S0002-9947-1978-0474240-2**[BC]**Louis Block and Ethan M. Coven,*𝜔-limit sets for maps of the interval*, Ergodic Theory Dynam. Systems**6**(1986), no. 3, 335–344. MR**863198**, 10.1017/S0143385700003539**[C]**W. A. Coppel,*Continuous maps of an interval*, Lecture Notes in Australian National University, 1984.**[CH]**Ethan M. Coven and G. A. Hedlund,*𝑃=𝑅 for maps of the interval*, Proc. Amer. Math. Soc.**79**(1980), no. 2, 316–318. MR**565362**, 10.1090/S0002-9939-1980-0565362-0**[CN]**Ethan M. Coven and Zbigniew Nitecki,*Nonwandering sets of the powers of maps of the interval*, Ergodic Theory Dynamical Systems**1**(1981), no. 1, 9–31. MR**627784****[N]**Zbigniew Nitecki,*Periodic and limit orbits and the depth of the center for piecewise monotone interval maps*, Proc. Amer. Math. Soc.**80**(1980), no. 3, 511–514. MR**581016**, 10.1090/S0002-9939-1980-0581016-9**[S]**A. N. Sharkovsky,*On some properties of discrete dynamical systems*, Iteration theory and its applications (Toulouse, 1982) Colloq. Internat. CNRS, vol. 332, CNRS, Paris, 1982, pp. 153–158. MR**805187****[XI]**Jin Cheng Xiong,*The attracting centre of a continuous self-map of the interval*, Ergodic Theory Dynam. Systems**8**(1988), no. 2, 205–213. MR**951269****[X2]**-,*The closure of periodic points of a piecewise monotone map of the interval*, preprint.**[X3]**Jin Cheng Xiong,*The periods of periodic points of continuous self-maps of the interval whose recurrent points form a closed set*, J. China Univ. Sci. Tech.**13**(1983), no. 1, 134–135. MR**701790****[X4]**Jin Cheng Xiong,*Ω(𝑓\midΩ(𝑓))=\overline{𝑃(𝑓)} for every continuous self-map 𝑓 of the interval*, Kexue Tongbao (English Ed.)**28**(1983), no. 1, 21–23. MR**740485****[X5]**Jin Cheng Xiong,*Set of almost periodic points of a continuous self-map of an interval*, Acta Math. Sinica (N.S.)**2**(1986), no. 1, 73–77. MR**877371**, 10.1007/BF02568524**[Y]**Lai Sang Young,*A closing lemma on the interval*, Invent. Math.**54**(1979), no. 2, 179–187. MR**550182**, 10.1007/BF01408935

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
58F20,
26A18,
58F08

Retrieve articles in all journals with MSC: 58F20, 26A18, 58F08

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1992-1079896-4

Keywords:
Chain recurrent point,
nonwandering point,
recurrent point,
turning point,
-limit point,
-limit point,
-limit point

Article copyright:
© Copyright 1992
American Mathematical Society