Global holomorphic solutions of a generalization of the Schröder equation
We discuss global holomorphic solutions of a generalization of the Schröder equation. Necessary and sufficient conditions are given for the equation to have a holomorphic solution on a set extended from the Fatou component of an attracting or indifferent fixed point of a known function. Then we extend the result from a fixed point to a periodic point. The existence of holomorphic solutions is given by considering the complex dynamics of a known function. The continuation of holomorphic solutions is studied by investigating relations between the natural boundary of solutions and the Julia set of the known function.