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Dynamical interpretation of Schröder’s equation. Its consequences. (English) Zbl 1139.39314

Summary: This text deals with the domain of existence of the solution of Schröder’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 identify 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.

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
39B12 Iteration theory, iterative and composite equations
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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