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.

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

Article copyright:
© Copyright 1978
American Mathematical Society