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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
MathSciNet review: 509258
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0509258-X
PII: S 0002-9939(1978)0509258-X
Article copyright: © Copyright 1978 American Mathematical Society