The Logistic Function, Part II: Topological Conjugacy

Abstract

We return now to the analysis of the logistic function. Our goal is to prove that if $r > 2 + \sqrt 5$, then h_r(x) = rx(1 − x) is chaotic on Λ. We recall that Λ is the set of all numbers in [0, 1] which remain in [0, 1] under iteration of h. That is, Λ = {x | hⁿ(x) is in [0, 1] for all n}. By Theorem 9.20, it is sufficient to show that the periodic points of h are dense in Λ and that h is topologically transitive on Λ. We have already shown that the periodic points of h are dense on Λ in exercise 8.4. Unfortunately, proving that h is topologically transitive on Λ directly from the definition is a relatively difficult task. Consequently, we will show instead that the dynamics of h on Λ are the same as the dynamics of σ on Σ₂. Mathematically speaking, we say that h on Λ is topologically conjugate to σ on Σ₂.