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 | 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]**L. Block,*Continuous maps of the interval with finite nonwandering set*, Trans. Amer. Math. Soc.**240**(1978), 221-230. MR**0474240 (57:13887)****[BC]**L. Block and E. M. Coven,*-limit sets for maps of the interval*, Ergodic Theory Dynamical Systems**6**(1986), 335-344. MR**863198 (88a:58165)****[C]**W. A. Coppel,*Continuous maps of an interval*, Lecture Notes in Australian National University, 1984.**[CH]**E. M. Coven and G. H. Hedlund,*for maps of the interval*, Proc. Amer. Math. Soc.**79**(1980), 316-318. MR**565362 (81b:54042)****[CN]**E. M. Coven and N. Nitecki,*Nonwandering sets of the powers of maps of the interval Ergodic*Theory and Dynamical Systems**1**(1981), 9-13. MR**627784 (82m:58043)****[N]**Z. Nitecki,*Periodic and limit orbits and the depth of the centre for piecewise monotone interval maps*, Proc. Amer. Math. Soc.**80**(1980), 511-514. MR**581016 (81j:58068)****[S]**A. N. Sarkovskii,*On some properties of discrete dynamical systems*, Proc. Internat. Colloq. on Iteration Theory and its Appl. (Toulouse, 1982), Univ. Paul Sabatier, pp. 153-158. MR**805187 (86k:58058)****[XI]**J.-C. Xiong,*The attracting center of a continuous self-map of the interval*, Ergodic Theory Dynamical Systems**8**(1988), 205-213. MR**951269 (90b:58131)****[X2]**-,*The closure of periodic points of a piecewise monotone map of the interval*, preprint.**[X3]**-,*The perioids of periodic points of continuous self-maps of the interval whose recurrent points form a closed set*, J. China Univ. Sci. Tech.**13**(1983), 134-135. MR**701790 (84h:58124b)****[X4]**-,*for every continuous self map**of the interval*, Kexue Tongbao (English version)**28**(1983), 21-23. MR**740485 (85m:58150)****[X5]**-,*Set of almost periodic points of a continuous self-map of an interval*, Acta Math. Sinica (N.S.)**2**(1986), 73-77. MR**877371 (88d:58093)****[Y]**L. Young,*A closing lemma on the interval*, Invent. Math.**54**(1979), 179-187. MR**550182 (80k:58084)**

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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,
-limit point,
-limit point,
-limit point

Article copyright:
© Copyright 1992
American Mathematical Society