The Logistic Function Part IV: Symbolic Dynamics

Abstract

We begin this chapter with a discussion of metric spaces and symbolic dynamics. As we proceed through the discussion, it may seem that the dynamics of the shift map on symbol space is an odd place to look for a deeper understanding of the logistic map, but we shall see that it is precisely the tool we need. In particular, we culminate the chapter by using symbolic dynamics to show that when $r > 2 + \sqrt 5$, then there is a point in the set Λ whose orbit under iteration of h _r (x) = rx(1 − x) is dense in Λ.