Formal power series solutions of Schröder’s equation

In 1884, Konigs showed that when \(\varphi\)(z) is an analytic self-map of the unit disk fixing the origin, with 0 1. Under some natural hypotheses, they gave necessary and sufficient conditions for the existence of an analytic solution σ satisfying σ′(0) = I when \(\varphi\)′(0) is diagonalizable. In this paper, the problem when \(\varphi\)′(0) is not diagonalizable is considered. Both \(\varphi\)(z) and σ(z) will be regarded as vectors of purely formal power series, and it will not be assumed that \(\varphi\)(z) is analytic or that the series for \(\varphi\)(z) or σ(z) converge. Nevertheless, because of a process developed by Cowen and MacCluer, if the given \(\varphi\)(z) represents a map of the unit ball into itself of an appropriate form, then the results of this paper can be used to produce solutions of Schroder’s equation that are convergent power series, or (sometimes) to show that no such solution exists. A method of matrix completion is used.