Simple periodic orbits of mappings of the initial

Author:
Louis Block

Journal:
Trans. Amer. Math. Soc. **254** (1979), 391-398

MSC:
Primary 58F20; Secondary 28D20, 54H20

MathSciNet review:
539925

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Abstract: Let *f* be a continuous map of a closed, bounded interval into itself. A criterion is given to determine whether or not *f* has a periodic point whose period is not a power of 2, which just depends on the periodic orbits of *f* whose period is a power of 2. Also, a lower bound for the topological entropy of *f* is obtained.

**[1]**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**[2]**Rufus Bowen and John Franks,*The periodic points of maps of the disk and the interval*, Topology**15**(1976), no. 4, 337–342. MR**0431282****[3]**L. Jonker and D. Rand,*A lower bound for the entropy of certain maps of the unit interval*(preprint).**[4]**Tien Yien Li and James A. Yorke,*Period three implies chaos*, Amer. Math. Monthly**82**(1975), no. 10, 985–992. MR**0385028****[5]**R. M. May,*Simple mathematical models with very complicated dynamics*, Nature**261**(1976), 459-467.**[6]**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****[7]**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/S0002-9947-1979-0539925-9

Article copyright:
© Copyright 1979
American Mathematical Society