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}, Aⁿ = {a₀∩f⁻¹a₁−…f⁻⁽ⁿ⁻¹⁾a_n−1 : a₀,a₁,…,a_n−1∊A}, ${\text{h}}\left( {{\text{f,}}\,{\text{A}}} \right) = \mathop {\lim }\limits_{{\text{n}} \to {{\alpha }}} \frac{1}{{\text{n}}}\log \,{\text{N}}\,\left( {{\text{A}}^{\text{n}} } \right)$