The chain recurrent set for maps of the circle

Block, Louis; Franke, John E.

doi:10.1090/S0002-9939-1984-0760951-6

The chain recurrent set for maps of the circle
HTML articles powered by AMS MathViewer

by Louis Block and John E. Franke PDF

Proc. Amer. Math. Soc. 92 (1984), 597-603 Request permission

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.

References

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, DOI 10.1017/S014338570000211X
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, DOI 10.1090/S0002-9939-1983-0687650-2
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

Continuous maps of the interval with a closed set of periodic points

John E. Franke and James F. Selgrade, Hyperbolicity and chain recurrence, J. Differential Equations 26 (1977), no. 1, 27–36. MR 467834, DOI 10.1016/0022-0396(77)90096-1
Zbigniew Nitecki, Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971. MR 0649788
Zbigniew Nitecki, Explosions in completely unstable flows. I. Preventing explosions, Trans. Amer. Math. Soc. 245 (1978), 43–61. MR 511399, DOI 10.1090/S0002-9947-1978-0511399-2
Zbigniew Nitecki, Maps of the interval with closed periodic set, Proc. Amer. Math. Soc. 85 (1982), no. 3, 451–456. MR 656122, DOI 10.1090/S0002-9939-1982-0656122-2
Z. Nitecki and M. Shub, Filtrations, decompositions, and explosions, Amer. J. Math. 97 (1975), no. 4, 1029–1047. MR 394762, DOI 10.2307/2373686
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
Ken Sawada, On the iterations of diffeomorphisms without $C^{0}-\Omega$-explosions: an example, Proc. Amer. Math. Soc. 79 (1980), no. 1, 110–112. MR 560595, DOI 10.1090/S0002-9939-1980-0560595-1
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
M. Shub and S. Smale, Beyond hyperbolicity, Ann. of Math. (2) 96 (1972), 587–591. MR 312001, DOI 10.2307/1970826
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