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

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

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]**L. Block,*Simple periodic orbits of mappings of the interval*, Trans. Amer. Math. Soc.**254**(1979), 391-398. MR**539925 (80m:58031)****[2]**T. Li and J. A. Yorke,*Period three implies chaos*, Amer. Math. Monthly**82**(1975), 985-992. MR**0385028 (52:5898)****[3]**R. M. May,*Simple mathematical models with very complicated dynamics*, Nature**261**(1976), 459-467.**[4]**A. N. Šarkovskii,*Coexistence of cycles of a continuous map of a line into itself*, Ukrain. Mat. Ž.**16**(1964), 61-71. 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**0445556 (56:3894)**

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Additional Information

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