Homoclinic points of mappings of the interval
HTML articles powered by AMS MathViewer
- by Louis Block
- Proc. Amer. Math. Soc. 72 (1978), 576-580
- DOI: https://doi.org/10.1090/S0002-9939-1978-0509258-X
- PDF | Request permission
Abstract:
Let f be a continuous map of a closed interval I into itself. A point $x \in I$ is called a homoclinic point of f if there is a peridoic point p of f such that $x \ne p,x$ is in the unstable manifold of p, and p is in the orbit of x under ${f^n}$, 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 ${f^n}(X) = X$ and there is a topological semiconjugacy of ${f^n}:X \to X$ onto the full (one-sided) shift on two symbols.References
- Louis Block, Continuous maps of the interval with finite nonwandering set, Trans. Amer. Math. Soc. 240 (1978), 221–230. MR 474240, DOI 10.1090/S0002-9947-1978-0474240-2
- 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 467841, DOI 10.1090/S0002-9939-1977-0467841-3
- Rufus Bowen and John Franks, The periodic points of maps of the disk and the interval, Topology 15 (1976), no. 4, 337–342. MR 431282, DOI 10.1016/0040-9383(76)90026-4
- 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
- S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
- 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 445556
Bibliographic Information
- © Copyright 1978 American Mathematical Society
- 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