ABSTRACT
This chapter introduces one of the most powerful tools for understanding the chaotic behavior of dynamical systems, symbolic dynamics. It introduces a new level of abstraction involving a "space" of sequences and a mapping on this space that will later serve as a model for the quadratic maps. A homeomorphism is a one-to-one, onto, and continuous function with continuous inverse. If researchers have a homeomorphism between two sets, these sets are said to be homeomorphic. Homeomorphism allows for sets to be distorted, as, for example, two finite closed intervals are homeomorphic, even though their lengths differ. But an open interval and a closed interval are never homeomorphic. After winning the Fields Medal in 1966 for resolving the Poincare Conjecture, Stephen Smale turned his attention to dynamical systems. In his famous "horseshoe" example, Smale showed that very simple dynamical systems could behave quite chaotically. Moreover, he introduced the use of symbolic dynamics to help understand this chaotic behavior.
