Iteration, Iterative Roots and Iterative Equations

Iteration exists extensively in the nature. Iteration of a homeomorphism generates a dynamical system. To embed such a homeomorphism into a flow we need to define fractional iteration and find iterative roots. The problem of iterative roots is to discuss iterative equations. This is a way to lead us to explore the mathematics of iteration. Iteration is a basic concept in the theory of dynamical systems. Over a hundred years ago E. Schröder, N. H. Abel, and C. Babbage studied iteration and obtained many nice results [20, 1, 4]. With iteration we trace where an object will go under the action of a dynamical system. On the other hand, we are also concerned with the course between two succesive states of iteration, inserting data to preserve iteration and embedding a discrete dynamical system into a flow. In section 1 we introduce some basic knowledges about iteration. Then we give some concepts, ideas, results and problems for iterative roots and iterative equations respectively in section 2 and 3.