Homoclinic points of mappings of the interval

Author:
Louis Block

Journal:
Proc. Amer. Math. Soc. **72** (1978), 576-580

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

DOI:
https://doi.org/10.1090/S0002-9939-1978-0509258-X

MathSciNet review:
509258

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Abstract: Let *f* be a continuous map of a closed interval *I* into itself. A point is called a homoclinic point of *f* if there is a peridoic point *p* of *f* such that is in the unstable manifold of *p*, and *p* is in the orbit of *x* under , where *n* is the period of *p*. It is shown that *f* has a homoclinic point if and only if *f* has a periodic point whose period is not a power of 2. Furthermore, in this case, there is a subset *X* of *I* and a positive integer *n*, such that and there is a topological semiconjugacy of onto the full (one-sided) shift on two symbols.

**[1]**Louis Block,*Continuous maps of the interval with finite nonwandering set*, Trans. Amer. Math. Soc.**240**(1978), 221–230. MR**0474240**, https://doi.org/10.1090/S0002-9947-1978-0474240-2**[2]**Louis Block,*Mappings of the interval with finitely many periodic points have zero entropy*, Proc. Amer. Math. Soc.**67**(1977), no. 2, 357–360. MR**0467841**, https://doi.org/10.1090/S0002-9939-1977-0467841-3**[3]**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**, https://doi.org/10.1016/0040-9383(76)90026-4**[4]**O. M. Šarkovs′kiĭ,*Co-existence of cycles of a continuous mapping of the line into itself*, Ukrain. Mat. Ž.**16**(1964), 61–71 (Russian, with English summary). MR**0159905****[5]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747–817. MR**0228014**, https://doi.org/10.1090/S0002-9904-1967-11798-1**[6]**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/S0002-9939-1978-0509258-X

Article copyright:
© Copyright 1978
American Mathematical Society