Sets of recurrent points of continuous maps of the interval

Author:
Jin Cheng Xiong

Journal:
Proc. Amer. Math. Soc. **95** (1985), 491-494

MSC:
Primary 58F20; Secondary 54H20

MathSciNet review:
806094

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a continuous map of the interval the following conditions are equivalent: (1) the period of every periodic point is a power of 2, (2) , and (3) is countable, where denotes the set of recurrent points. is the closure of , and (or ) is the right-side closure (left-side closure, respectively) of .

**[1]**Jin Cheng Xiong,*The periods of periodic points of continuous self-maps of the interval whose recurrent points form a closed set*, J. China Univ. Sci. Tech.**13**(1983), no. 1, 134–135. MR**701790****[2]**A. M. Blokh,*The asymptotic behaviour of one-dimensional system*, Uspehki Mat. Nauk**37**(1982), no. 1, 137-138.**[3]**Hsin Chu and Jin Cheng Xiong,*A counterexample in dynamical systems of the interval*, Proc. Amer. Math. Soc.**97**(1986), no. 2, 361–366. MR**835899**, 10.1090/S0002-9939-1986-0835899-0**[4]**P. Erdös and A. H. Stone,*Some remarks on almost periodic transformations*, Bull. Amer. Math. Soc.**51**(1945), 126–130. MR**0011437**, 10.1090/S0002-9904-1945-08289-5**[5]**A. N. Sarkovskii,*Nonwandering points and the centre of a continuous map of the line*int*itself*, Dorporidi Akad. Nauk. Ukrain, RSR Ser. A 1964, pp. 865-868.**[6]**Ethan M. Coven and G. A. Hedlund,*𝑃=𝑅 for maps of the interval*, Proc. Amer. Math. Soc.**79**(1980), no. 2, 316–318. MR**565362**, 10.1090/S0002-9939-1980-0565362-0**[7]**Jin Cheng Xiong,*Ω(𝑓\midΩ(𝑓))=𝑃(𝑓) for a continuous self-mapping 𝑓 of an interval*, Kexue Tongbao (Chinese)**27**(1982), no. 9, 513–514 (Chinese). MR**732267****[8]**Tien Yien Li, Michał Misiurewicz, Giulio Pianigiani, and James A. Yorke,*Odd chaos*, Phys. Lett. A**87**(1981/82), no. 6, 271–273. MR**643455**, 10.1016/0375-9601(82)90692-2**[9]**Louis Block,*Homoclinic points of mappings of the interval*, Proc. Amer. Math. Soc.**72**(1978), no. 3, 576–580. MR**509258**, 10.1090/S0002-9939-1978-0509258-X

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
58F20,
54H20

Retrieve articles in all journals with MSC: 58F20, 54H20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1985-0806094-5

Article copyright:
© Copyright 1985
American Mathematical Society