The chain recurrent set for maps of the circle

Authors:
Louis Block and John E. Franke

Journal:
Proc. Amer. Math. Soc. **92** (1984), 597-603

MSC:
Primary 58F12; Secondary 54H20, 58F20

MathSciNet review:
760951

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Abstract | References | Similar Articles | Additional Information

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**, 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**, 10.1090/S0002-9939-1983-0687650-2**[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****[4]**V. V. Fedorenko and A. N. Šarkovskii,*Continuous maps of the interval with a closed set of periodic points*, Studies of Differential and Differential-Delay Equations, Kiev, 1980, pp. 137-145. (Russian)**[5]**John E. Franke and James F. Selgrade,*Hyperbolicity and chain recurrence*, J. Differential Equations**26**(1977), no. 1, 27–36. MR**0467834****[6]**Zbigniew Nitecki,*Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms*, The M.I.T. Press, Cambridge, Mass.-London, 1971. MR**0649788****[7]**Zbigniew Nitecki,*Explosions in completely unstable flows. I. Preventing explosions*, Trans. Amer. Math. Soc.**245**(1978), 43–61. MR**511399**, 10.1090/S0002-9947-1978-0511399-2**[8]**Zbigniew Nitecki,*Maps of the interval with closed periodic set*, Proc. Amer. Math. Soc.**85**(1982), no. 3, 451–456. MR**656122**, 10.1090/S0002-9939-1982-0656122-2**[9]**Z. Nitecki and M. Shub,*Filtrations, decompositions, and explosions*, Amer. J. Math.**97**(1975), no. 4, 1029–1047. MR**0394762****[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****[11]**Ken Sawada,*On the iterations of diffeomorphisms without 𝐶⁰-Ω-explosions: an example*, Proc. Amer. Math. Soc.**79**(1980), no. 1, 110–112. MR**560595**, 10.1090/S0002-9939-1980-0560595-1**[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****[13]**M. Shub and S. Smale,*Beyond hyperbolicity*, Ann. of Math. (2)**96**(1972), 587–591. MR**0312001****[14]**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**

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DOI:
https://doi.org/10.1090/S0002-9939-1984-0760951-6

Article copyright:
© Copyright 1984
American Mathematical Society