## ABSTRACT

There are many kinds of problems in science and mathematics that involve iteration. Iteration means to repeat a process over and over. In dynamics, the process that is repeated is the application of a function. This chapter considers only functions of one real variable as encountered in elementary calculus. To iterate a function means to evaluate the function over and over, using the output of the previous application as the input for the next. This is the same process as typing a number into a calculator, then repeatedly striking one of the function keys such as “sin” or “cos.” There are many different kinds of orbits in a typical dynamical system. Undoubtedly the most important kind of orbit is a fixed point. One of the major discoveries in mathematics is that many simple functions—such as quadratic functions of a real variable—may have many orbits of incredible complexity.