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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Homoclinic points of mappings of the interval
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by Louis Block
Proc. Amer. Math. Soc. 72 (1978), 576-580
DOI: https://doi.org/10.1090/S0002-9939-1978-0509258-X

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
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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