ABSTRACT
This chapter looks at a topological model of the process and uses symbolic dynamics to prove that the process does, in fact, lead to chaos. The doubling map and the logistic function both exhibit chaotic behavior on the interval. The fact that almost every point in the unit interval has 2 preimages somehow causes the interval to get so mixed up under iteration that the dynamics on this interval are chaotic. Smale was essentially interested in constructing an example of a chaotic dynamical that was both invertible and hyperbolic (that is, persistent under small perturbations). The chapter ends with a discussion of the Henon map, homoclinic tangles, and saddles.
