Relations between chain recurrent points and turning points on the interval
HTML articles powered by AMS MathViewer
- by Shi Hai Li PDF
- Proc. Amer. Math. Soc. 115 (1992), 265-270 Request permission
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.References
- Louis Block, Continuous maps of the interval with finite nonwandering set, Trans. Amer. Math. Soc. 240 (1978), 221–230. MR 474240, DOI 10.1090/S0002-9947-1978-0474240-2
- Louis Block and Ethan M. Coven, $\omega$-limit sets for maps of the interval, Ergodic Theory Dynam. Systems 6 (1986), no. 3, 335–344. MR 863198, DOI 10.1017/S0143385700003539 W. A. Coppel, Continuous maps of an interval, Lecture Notes in Australian National University, 1984.
- Ethan M. Coven and G. A. Hedlund, $\bar P=\bar R$ for maps of the interval, Proc. Amer. Math. Soc. 79 (1980), no. 2, 316–318. MR 565362, DOI 10.1090/S0002-9939-1980-0565362-0
- Ethan M. Coven and Zbigniew Nitecki, Nonwandering sets of the powers of maps of the interval, Ergodic Theory Dynam. Systems 1 (1981), no. 1, 9–31. MR 627784, DOI 10.1017/s0143385700001139
- 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, DOI 10.1090/S0002-9939-1980-0581016-9
- 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
- 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, DOI 10.1017/S0143385700004429 —, The closure of periodic points of a piecewise monotone map of the interval, preprint.
- Jin Cheng Xiong, Continuous self-maps of the closed interval whose periodic points form a closed set, J. China Univ. Sci. Tech. 11 (1981), no. 4, 14–23 (English, with Chinese summary). MR 701781
- Jin Cheng Xiong, $\Omega (f\mid \Omega (f))={\overline {P(f)}}$ for every continuous self-map $f$ of the interval, Kexue Tongbao (English Ed.) 28 (1983), no. 1, 21–23. MR 740485
- 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, DOI 10.1007/BF02568524
- Lai Sang Young, A closing lemma on the interval, Invent. Math. 54 (1979), no. 2, 179–187. MR 550182, DOI 10.1007/BF01408935
Additional Information
- © Copyright 1992 American Mathematical Society
- 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