The chain recurrent set for maps of the circle
Authors:
Louis Block and John E. Franke
Journal:
Proc. Amer. Math. Soc. 92 (1984), 597603
MSC:
Primary 58F12; Secondary 54H20, 58F20
MathSciNet review:
760951
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Abstract: For a continuous map of the circle to itself we give necessary and sufficient conditions for the chain recurrent set to be precisely the set of periodic points. We also examine the possible types of examples which can occur, where the set of periodic points is closed and nonempty, but there are nonperiodic, chain recurrent points.
 [1]
Louis
Block, Ethan
Coven, Irene
Mulvey, and Zbigniew
Nitecki, Homoclinic and nonwandering points for maps of the
circle, Ergodic Theory Dynam. Systems 3 (1983),
no. 4, 521–532. MR 753920
(86b:58101), http://dx.doi.org/10.1017/S014338570000211X
 [2]
Louis
Block and John
E. Franke, The chain recurrent set for maps of
the interval, Proc. Amer. Math. Soc.
87 (1983), no. 4,
723–727. MR
687650 (84j:58103), http://dx.doi.org/10.1090/S00029939198306876502
 [3]
Charles
Conley, Isolated invariant sets and the Morse index, CBMS
Regional Conference Series in Mathematics, vol. 38, American
Mathematical Society, Providence, R.I., 1978. MR 511133
(80c:58009)
 [4]
V. V. Fedorenko and A. N. Šarkovskii, Continuous maps of the interval with a closed set of periodic points, Studies of Differential and DifferentialDelay Equations, Kiev, 1980, pp. 137145. (Russian)
 [5]
John
E. Franke and James
F. Selgrade, Hyperbolicity and chain recurrence, J.
Differential Equations 26 (1977), no. 1, 27–36.
MR
0467834 (57 #7685)
 [6]
Zbigniew
Nitecki, Differentiable dynamics. An introduction to the orbit
structure of diffeomorphisms, The M.I.T. Press, Cambridge,
Mass.London, 1971. MR 0649788
(58 #31210)
 [7]
Zbigniew
Nitecki, Explosions in completely unstable
flows. I. Preventing explosions, Trans. Amer.
Math. Soc. 245
(1978), 43–61. MR 511399
(81e:58030), http://dx.doi.org/10.1090/S00029947197805113992
 [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]
Z.
Nitecki and M.
Shub, Filtrations, decompositions, and explosions, Amer. J.
Math. 97 (1975), no. 4, 1029–1047. MR 0394762
(52 #15561)
 [10]
O.
M. Šarkovs′kiĭ, On cycles and the structure of
a continuous mapping, Ukrain. Mat. Ž. 17
(1965), no. 3, 104–111 (Russian). MR 0186757
(32 #4213)
 [11]
Ken
Sawada, On the iterations of diffeomorphisms
without 𝐶⁰Ωexplosions: an example, Proc. Amer. Math. Soc. 79 (1980), no. 1, 110–112. MR 560595
(81h:58055), http://dx.doi.org/10.1090/S00029939198005605951
 [12]
Michael
Shub, Stabilité globale des systèmes dynamiques,
Astérisque, vol. 56, Société Mathématique
de France, Paris, 1978 (French). With an English preface and summary. MR 513592
(80c:58015)
 [13]
M.
Shub and S.
Smale, Beyond hyperbolicity, Ann. of Math. (2)
96 (1972), 587–591. MR 0312001
(47 #563)
 [14]
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, E. Coven, I. Mulvey and Z. Nitecki, Homoclinic and nonwandering point for maps of the circle, Ergodic Theory Dynamical Systems (to appear). MR 753920 (86b:58101)
 [2]
 L. Block and J. Franke, The chain recurrent set for maps of the interval, Proc. Amer. Math. Soc. 87 (1983), 723727. MR 687650 (84j:58103)
 [3]
 C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., no. 38. Amer. Math. Soc., Providence, R.I., 1976. MR 511133 (80c:58009)
 [4]
 V. V. Fedorenko and A. N. Šarkovskii, Continuous maps of the interval with a closed set of periodic points, Studies of Differential and DifferentialDelay Equations, Kiev, 1980, pp. 137145. (Russian)
 [5]
 J. Franke and J. Selgrade, Hyperbolicity and chain recurrence, J. Differential Equations 26 (1977), 2736. MR 0467834 (57:7685)
 [6]
 Z. Nitecki, Differentiable dynamics, M.I.T. Press, Cambridge, Mass., 1971. MR 0649788 (58:31210)
 [7]
 , Explosions in completely unstable flows. I. preventing explosions, Trans. Amer. Math. Soc. 245 (1978), 4361. MR 511399 (81e:58030)
 [8]
 , Maps of the interval with closed period set, Proc. Amer. Math. Soc. 85 (1982), 451456. MR 656122 (83k:58067)
 [9]
 Z. Nitecki and M. Shub, Filtrations, decompositions, and explosions, Amer. J. Math. 97 (1976), 10291047. MR 0394762 (52:15561)
 [10]
 A. N. Šarkovskiĭ, On cycles and the structure of a continuous mapping, Ukrainian Math. J. 17 (1965), 104111. (Ukrainian) MR 0186757 (32:4213)
 [11]
 K. Sawada, On the iterations of diffeomorphisms without explosions: an example, Proc. Amer. Math. Soc. 79 (1980), 110112. MR 560595 (81h:58055)
 [12]
 M. Shub, Stabilitéglobal des systèmes dynamiques, Asterisque, vol. 56, Soc. Math. France, Paris, 1978. MR 513592 (80c:58015)
 [13]
 M. Shub and S. Smale, Beyond hyperbolicity, Ann. of Math. (2) 96 (1972), 587591. MR 0312001 (47:563)
 [14]
 J. C. Xiong, Continuous selfmaps of the closed interval whose periodic points form a closed set, J. China Univ. Sci. 11 (1981), 1423. MR 701781 (84h:58124a)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198407609516
PII:
S 00029939(1984)07609516
Article copyright:
© Copyright 1984
American Mathematical Society
