Dynamical interpretation of Schröder's equation. Its consequences

Abstract

This text deals with the domain of existence of the solution of Schroder’s equation, related to a two-dimensional real iteration process, defined by functions which do not satisfy the Cauchy-Riemann conditions. Its purpose is limited to the identification of the difficulties generated by the determination of this domain. When the Cauchy- Riemann conditions are verified the answer to this problem was given by Fatou at the beginning of the 20th century. The qualitative theory of dynamical systems permits to identity the difficulties which may be met, from the notion of immediate basin of an attractor (stable fixed point in our case), and the singular set generated by the iteration associated with Schröder’s equation.