The chain recurrent set for maps of the interval
Authors:
Louis Block and John E. Franke
Journal:
Proc. Amer. Math. Soc. 87 (1983), 723727
MSC:
Primary 58F22; Secondary 54H20
MathSciNet review:
687650
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a continuous map of a compact interval into itself. We show that if the set of periodic points of is a closed set then every chain recurrent point is periodic.
 [1]
Louis
Block, Mappings of the interval with finitely
many periodic points have zero entropy, Proc.
Amer. Math. Soc. 67 (1977), no. 2, 357–360. MR 0467841
(57 #7692), http://dx.doi.org/10.1090/S00029939197704678413
 [2]
Louis
Block, Homoclinic points of mappings of the
interval, Proc. Amer. Math. Soc.
72 (1978), no. 3,
576–580. MR
509258 (81m:58063), http://dx.doi.org/10.1090/S0002993919780509258X
 [3]
C. Conley, The gradient structure of a flow. I, IBM Research, RC 3932 (#17806), Yorktown Heights, N. Y., July 17, 1972.
 [4]
, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R. I., 1976.
 [5]
Ethan
M. Coven and G.
A. Hedlund, Continuous maps of the interval whose
periodic points form a closed set, Proc. Amer.
Math. Soc. 79 (1980), no. 1, 127–133. MR 560598
(81a:54042), http://dx.doi.org/10.1090/S00029939198005605987
 [6]
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
(82m:58043)
 [7]
Tien
Yien Li, Michał
Misiurewicz, Giulio
Pianigiani, and James
A. Yorke, Odd chaos, Phys. Lett. A 87
(1981/82), no. 6, 271–273. MR 643455
(83d:58058), http://dx.doi.org/10.1016/03759601(82)906922
 [8]
Zbigniew
Nitecki, Maps of the interval with closed
periodic set, Proc. Amer. Math. Soc.
85 (1982), no. 3,
451–456. MR
656122 (83k:58067), http://dx.doi.org/10.1090/S00029939198206561222
 [9]
Zbigniew
Nitecki, Topological dynamics on the interval, Ergodic theory
and dynamical systems, II (College Park, Md., 1979/1980), Progr. Math.,
vol. 21, Birkhäuser, Boston, Mass., 1982, pp. 1–73. MR 670074
(84g:54051)
 [10]
Jin
Cheng Xiong, Continuous selfmaps 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 (84h:58124a)
 [1]
 L. Block, Mappings of the interval with finitely many periodic points have zero entropy, Proc. Amer. Math. Soc. 67 (1977), 357360. MR 0467841 (57:7692)
 [2]
 , Homoclinic points of mappings of the interval, Proc. Amer. Math. Soc. 72 (1978). 576580. MR 509258 (81m:58063)
 [3]
 C. Conley, The gradient structure of a flow. I, IBM Research, RC 3932 (#17806), Yorktown Heights, N. Y., July 17, 1972.
 [4]
 , Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R. I., 1976.
 [5]
 E. M. Coven and G. A. Hedlund, Continuous maps of the interval whose periodic points form a closed set, Proc. Amer. Math. Soc. 79 (1980), 127133. MR 560598 (81a:54042)
 [6]
 E. M. Coven and Z. Nitecki, Nonwandering sets of the powers of maps of the interval, Ergodic Theory Dynamical Systems 1 (1981), 931. MR 627784 (82m:58043)
 [7]
 T. Li, M. Misiurewicz, G. Pianigiani and J. Yorke, Odd chaos, preprint. MR 643455 (83d:58058)
 [8]
 Z. Nitecki, Maps of the interval with closed periodic set, Proc. Amer. Math. Soc. 85 (1982), 451456. MR 656122 (83k:58067)
 [9]
 , Topological dynamics on the interval, Ergodic Theory and Dynamical Systems II (College Park. Md. 19791980), Progress in Math., vol. 21, Birkhauser, Boston, 1982, pp. 173. MR 670074 (84g:54051)
 [10]
 JinCheng Ziong, Continuous selfmaps of the closed interval whose periodic points form a closed set, J. China Univ. Sci. and Tech. 11 (1981), 1423. MR 701781 (84h:58124a)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
58F22,
54H20
Retrieve articles in all journals
with MSC:
58F22,
54H20
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198306876502
PII:
S 00029939(1983)06876502
Article copyright:
© Copyright 1983
American Mathematical Society
