Horseshoes for continuous mappings of an interval

Abstract

Let X be a compact Hausdorff space and let f: X → X be a continuous mapping. Let us recall the notion of topological entropy. For a cover A of X we put N(A) = min {Card B: B is a subcover of A}, $h(f,A) = \mathop {\lim }\limits_{n \to \infty } \tfrac{1}{n}\log N({A^n})$, ${A^n} = \{ {a_0} \cap {f^{ - 1}}{a_1} \cap ... \cap {f^{ - (n - 1)}}{a_{n -1}}:{a_0},{a_1},...,{a_{n - 1}} \in A\}$ n (the limit always exists and is not greater than log N(A)). The topological entropy of f is defined as

$$h(f) = \sup \{ h(f,A):A{\kern 1pt} {\kern 1pt} is{\kern 1pt} an{\kern 1pt} {\kern 1pt} open{\kern 1pt} {\kern 1pt} \,\operatorname{cov} er\,of\,X\} $$

.