1 - Iteration

Summary

Introduction

This chapter is intended to give a brief review of the basic notions in the theory of iteration which (with one exception) will be needed in what follows. It is one of our leading ideas, expressed also in the title of the book, to emphasize the fruitful and intriguing interplay between iteration and the theory of functional equations in a single variable. We feel that any attempt to divorce iteration from functional equation investigations would be an extremely, indeed totally, fruitless task. In the whole book we shall endeavour to make the reader believe this.

Some readers may find some serious omissions in this chapter but we have attempted to minimize the contents of weighty terminology and, on the other hand, we are very far from an aspiration to any kind of completeness. Basic to this chapter are the following questions: iteration sequences (splinters) and orbits, cycles, attractive fixed points and the domain of attraction. Fixed points play a distinguished role both in functional equations and in iteration theory. Some fixed-point theorems which will be useful in the sequel are also included in the present chapter. One may consider them as various generalizations of the Banach contraction principle in a complete metric space.