Functional equations and iteration theory

Research project P 14342 Functional equation and interation theory Ludwig REICH 08.05.2000 In this project we propose the study of several problems from the theory of fundamental equations, in particular about functional equations in iteration theory. The research of members of the Institute of Mathematics of the Karl- Franzens University in Graz through many years indicates that these problems are interesting and promising. Besides iteration theory in rings of formal power series the sources of these problems are applications, algebraic aspects of additive functions, commutable functions and functional equations of iterative type. Especially we are dealing with functional equations and group actions, which were motivated by the apparent motion of the ``mean sun``. Another point of interest are derivations of higher order, which form a special class of additive mappings. We want to characterize the decomposable derivations of order 2, determine defining sets of them, and investigate stability properties of derivation of order n. Furthermore we want to determine all additive functions which commute with a prescribed class of rational functions. Especially real or complex functions defined on the set of real or complex numbers will be investigated. A similar situation as the Schröder and the Preschröder equation should be studied for the linear equation in order to end up with prelinear equations. The next problems arise from iteration theory. We want to give explicit solutions of the translation equation in formal power series rings. Furthermore solutions of the Aczél-Jabotinsky differential equation in formal power series rings. Furthermore solutions of the Aczél-Jabotinsky differential equations in the higher dimensional case should be described. Then we are looking for a covariant embedding of the linear functional equation into a family of linear equations. This would lead to generalizations of the Schröder series. In addition to this, solutions of the inhomogeneous Cauchy functional equation in connection with the stability of additive functions should be investigated. And finally asymptotic formulas for the iterates of a function should be given.

Our aim in this project was to develop a mathematical theory for several types of functional equations related to iteration problems and group actions involving functional equations, as general as reasonable and as complete as possible. Functional equations are relations among functions (mappings) from which some of these functions (the "unknowns") have to be determined. There are numerous and important applications of functional equations in physics, information theory and, more recent ones, in economics and in social and behavioral sciences. Many problems in geometry can be expressed by functional equations, and important special functions are characterized as solutions of such equations. Iteration theory is the theory of the iterates (compositions) of a selfmapping of a set and its generalizations, like flows, and may be considered as part of the theory of dynamical systems. There is a close relation between iteration problems and certain important functional equations (as the translation equation and Babbage`s equation of the iterative roots). On the other side, methods and concepts from iteration theory are often applied to solve functional equations (e.g. linear equations, Schröder type equations, etc.) The first part of the project was devoted to solving the problem of covariant embeddings of a linear functional equation with respect to a given iteration group (flow). This is a na-tural generalization of the basic problem to embed a given mapping into a flow, where the role of the mapping is taken by a linear functional equation together with its set of solutions. This problem was studied in the setting of formal power series. A system of functional equations (the cocycle equations) played a crucial role. One of these cocycle equations appeared before in other problems (automata theory, theory of stochastic processes.) The second part of the project is concerned with the connections between group actions (well known from group theory, combinatorics, and geometry) with functional equations. We studied from this point of view the so called linear-affine functional equation (appearing in the theory of measurement and in economics) under very general assumptions, and the so called equation of the mean sun (from mathematical astronomy) in an abstract setting and in the situation of matrix groups, in particular the group of the three-dimensional rotations.