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Homoclinic and non-wandering points for maps of the circle

  • Louis Block (a1), Ethan Coven (a2), Irene Mulvey (a3) and Zbigniew Nitecki (a4)

Abstract

For continuous maps ƒ of the circle to itself, we show: (A) the set of nonwandering points of ƒ coincides with that of ƒn for every odd n; (B) ƒ has a horseshoe if and only if it has a non-wandering homoclinic point; (C) if the set of periodic points is closed and non-empty, then every non-wandering point is periodic.

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References

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[1]Block, L.. Homoclinic points of mappings of the interval. Proc. Amer. Math. Soc. 72 (1978), 576580.
[2]Block, L., Coven, E. M. & Nitecki, Z.. Minimizing topological entropy for maps of the circle. Ergod. Th. & Dynam. Sys. 1 (1981), 145149.
[3]Coven, E. M. & Nitecki, Z.. Non-wandering sets of the powers of maps of the interval. Ergod. Th. & Dynam. Sys. 1 (1981), 931.
[4]Fedorenko, V. V. & Sarkovskii, A. N.. Continuous mappings of the interval with closed sets of periodic points. In Introduction to Differential and Differential-difference Equations, pp. 137145. Kiev, 1980. (Russian)
[5]Misiurewicz, M.. Horseshoes for mappings of the interval. Bull. Acad. Polon. Sci. Ser. Sci. Math. 27 (1979), 167169.
[6]Mulvey, I.. Periodic, recurrent and non-wandering points for continuous maps of the circle. Ph.D. thesis. Wesleyan University, Middletown, Conn. (1982).
[7]Nitecki, Z.. Topological dynamics on the interval. In Ergodic Theory and Dynamical Systems II, pp. 173. College Park, Md., 19791980,
Progr. Math. vol. 21. Birkhaüser: Boston, 1982.
[8]Nitecki, Z.. Maps of the interval with closed periodic set. Proc. Amer. Math. Soc. 85 (1982), 451456.
[9]Sarkovskii, A. N.. On cycles and structure of continuous mappings. Ukrain. Mat. Z. 17 (1965), 104111. (Russian)
[10]Sarkovskii, A. N.. On the problem of isomorphism of dynamical systems. In Proceedings of the International Conference on Nonlinear Trajectories, vol. 2, pp. 541545. Kiev, 1970. (Russian)
[11]Xiong, J.-C.. Continuous self-maps of the closed interval whose periodic points form a closed set. J. China University of Science and Technology 11 (1981), 1423.

Homoclinic and non-wandering points for maps of the circle

  • Louis Block (a1), Ethan Coven (a2), Irene Mulvey (a3) and Zbigniew Nitecki (a4)

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