Formal power series solutions of Schröder's equation (Q2458386)

In 1884, \textit{G. Koenigs} [Ann. de l'Ec. Norm. Supplém. 1--41 (1884; JFM 16.0376.01)] showed that when \(\varphi(z)\) is an analytic self-map of the unit disk fixing the origin, with \(0<| \varphi'(0)| <1\), then Schröder's functional equation, \(\sigma\circ \varphi(z)=\varphi'(0)\sigma(z)\), can be solved for a unique analytic function \(\sigma(z)\) in the disk with \(\sigma'(0)=1\). In 2003, \textit{C. C. Cowen} and \textit{B. D. MacCluer} [Taiwanese J. Math. 7, No. 1, 129--154 (2003; Zbl 1033.32011)] considered an analogue of Schröder's equation in the unit ball of \(\mathbb C^N\) for \(N>1\), and gave necessary and sufficient conditions for the existence of an analytic solution \(\sigma\) satisfying \(\sigma'(0)=I\), when \(\sigma'(0)\) is diagonalizable. In the present paper, the problem when \(\sigma'(0)\) is not diagonalizable is considered; moreover \(\varphi(z)\) and \(\sigma(z)\) will be regarded as vectors of purely formal power series. A method of matrix completion is used.