Continuous maps of the interval whose periodic points form a closed set

Coven, Ethan M.; Hedlund, G. A.

doi:10.1090/S0002-9939-1980-0560598-7

Continuous maps of the interval whose periodic points form a closed set
HTML articles powered by AMS MathViewer

by Ethan M. Coven and G. A. Hedlund

Proc. Amer. Math. Soc. 79 (1980), 127-133

DOI: https://doi.org/10.1090/S0002-9939-1980-0560598-7

PDF | Request permission

Abstract:

We show that for a continuous map of a closed interval to itself, if the set of periodic points is closed, then every recurrent point is periodic. If, furthermore, the set of least periods of the periodic points is finite, then every nonwandering point is periodic. This answers a question of L. Block [Proc. Amer. Math. Soc. 67 (1977), 357-360].

References

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 467841, DOI 10.1090/S0002-9939-1977-0467841-3
Louis Block, Continuous maps of the interval with finite nonwandering set, Trans. Amer. Math. Soc. 240 (1978), 221–230. MR 474240, DOI 10.1090/S0002-9947-1978-0474240-2
P. Erdös and A. H. Stone, Some remarks on almost periodic transformations, Bull. Amer. Math. Soc. 51 (1945), 126–130. MR 11437, DOI 10.1090/S0002-9904-1945-08289-5
O. M. Šarkovs′kiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Ž. 16 (1964), 61–71 (Russian, with English summary). MR 0159905
P. Štefan, A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977), no. 3, 237–248. MR 445556