Stability of periodic orbits in the theorem of Šarkovskii

Author:
Louis Block

Journal:
Proc. Amer. Math. Soc. **81** (1981), 333-336

MSC:
Primary 58F20; Secondary 28D05

MathSciNet review:
593484

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Abstract: Let be a continuous map of a closed, bounded interval into itself. It is shown that the conclusion of the theorem of Sarkovskii holds for perturbations of . In other words, if has a periodic point of period , and is a continuous map close to , then has periodic points of certain periods.

**[1]**Louis Block,*Simple periodic orbits of mappings of the interval*, Trans. Amer. Math. Soc.**254**(1979), 391–398. MR**539925**, 10.1090/S0002-9947-1979-0539925-9**[2]**Tien Yien Li and James A. Yorke,*Period three implies chaos*, Amer. Math. Monthly**82**(1975), no. 10, 985–992. MR**0385028****[3]**R. M. May,*Simple mathematical models with very complicated dynamics*, Nature**261**(1976), 459-467.**[4]**O. M. Šarkovs′kiĭ,*Co-existence of cycles of a continuous mapping of the line into itself*, Ukrain. Mat. Z.**16**(1964), 61–71 (Russian, with English summary). MR**0159905****[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), no. 3, 237–248. MR**0445556**

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DOI:
https://doi.org/10.1090/S0002-9939-1981-0593484-8

Keywords:
Periodic point,
period of a periodic point

Article copyright:
© Copyright 1981
American Mathematical Society