ABSTRACT
The theme of the chapter is complicated dynamics or chaos of maps on the line. The first section presents a theorem of Sharkovskii; it proves for certain n and k, that the existence of a periodic orbit of period n forces other orbits of period k. This theorem is not exactly about complicated dynamics, but it does show that if a map f on the line has a period which is not equal to a power of 2, then f has infinitely many different periods. The existence of infinitely many different periods is an indication of the complexity of such a map. This theorem is not used later in the book but is of interest in itself, especially since it is proved using mainly the Intermediate Value Theorem and combinatorial bookkeeping. Sharkovskii’s Theorem motivates the treatment of subshifts of finite type which is given in the next section. A subshift of finite type is determined by specifying which transitions are allowed between a finite set of states. Given such a system, it is easy to determine the periods which occur and other aspects of the dynamics. These systems are generalizations of the symbolic dynamics which we introduced for the quadratic map in Chapter II. In Chapter VIII and X, we give further examples of nonlinear dynamical systems which are conjugate to subshifts of finite type. Because we can analyze the subshift of finite type, we determine the complexity of the dynamics of the nonlinear map.
