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

Authors:
Ethan M. Coven and G. A. Hedlund

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

MSC:
Primary 54H20; Secondary 58F20

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

MathSciNet review:
560598

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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].

**[1]**L. Block,*Mappings of the interval with finitely many periodic points have zero entropy*, Proc. Amer. Math. Soc.**67**(1977), 357-360. MR**57**#7692. MR**0467841 (57:7692)****[2]**-,*Continuous maps of the interval with finite nonwandering set*, Trans. Amer. Math. Soc.**240**(1978), 221-230. MR**57**# 13887. MR**0474240 (57:13887)****[3]**P. Erdös and A. H. Stone,*Some remarks on almost periodic transformations*, Bull. Amer. Math. Soc.**51**(1945), 126-130. MR**6**, 165. MR**0011437 (6:165b)****[4]**A. N. Šarkovskiĭ,*Co-existence of cycles of a continuous mapping of the line into itself*, Ukrain. Mat. Ž.**16**(1964), 61-71. (Russian, English summary) MR**28**#3121. MR**0159905 (28:3121)****[5]**P. Štefan,*A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line*, Comm. Math. Phys.**54**(1977), 237-248. MR**56**#3898. MR**0445556 (56:3894)**

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

Article copyright:
© Copyright 1980
American Mathematical Society