Functional Equations related to the Iteration of Functions

Certain systems of functional equations related to the iteration of functions with a fixed point are considered. We construct smooth solutions in terms of expansions about a fixed point. In a particular example taken from an intuitive geometric situation the solution is obtained explicitly as a convergent Taylor series. Particular attention is given to the question of selecting distinguished solutions from a continuum of possible solutions. This classical topic is presented in a transparent way by consistently using compositional notation. The method described may be applied in similar situations, e.g. for handling iterations arising in discrete dynamical systems.