ABSTRACT
The Schwarzian derivative is one of the stranger tools in dynamics. Although this derivative has a venerable history in the field of complex analysis, but it was only introduced into the study of dynamical systems in 1978. Functions with negative Schwarzian derivatives have very interesting dynamical properties that simplify their analysis. The Schwarzian derivative of a function F is SF(x)=F‴(x)F'(x)-32(F″(x)F'(x))2. Many functions have negative Schwarzian derivatives. Many polynomials have the negative Schwarzian derivatives. The main reason for the importance of negative Schwarzian derivatives is the fact that this property is preserved by composition of functions and consequently by iteration. This chapter investigates how the assumption of negative Schwarzian derivative severely limits the kinds of dynamical behavior that may occur. It examines that each attracting periodic orbit of such a function must attract at least one critical point of the function.
