Homoclinic points of mappings of the interval

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.