A Solution to Schroeder's Equation in Several Variables

TL;DR

This paper extends the understanding of Schroeder's equation in several complex variables, providing necessary and sufficient conditions for solutions without the diagonalizability assumption, and analyzing solutions for higher powers.

Contribution

It offers a complete characterization of solutions to Schroeder's equation in several variables without requiring diagonalizability of the derivative at 0.

Findings

Solutions exist for F irc \u03a6 = t 0 without diagonalizability.

For k > 1, solutions are linearly independent but not injective near 0.

Formal power series solutions correspond to actual analytic functions.

Abstract

Let \phi be a self-map of B^n, the unit ball in C^n, fixing 0, and having full-rank at 0. If \phi (0)= 0, Koenigs proved in 1884 that in the well- known case n = 1, Schroeder's equation, f \circ \phi = \phi '(0) f has a solution f, which is bijective near 0 precisely when \phi '(0) \neq 0. In 2003, Cowen and MacCluer formulated the analogous problem in C^n (for a non-negative integer n) by defining Schroeder's equation in several variables as F \circ \phi = \phi '(0)F and giving appropriate assumptions on \phi . The 2003 Cowen and MacCluer paper also provides necessary and sufficient conditions for an analytic solution, F taking values in C^n and having full-rank near 0 under the additional assumption that \phi '(0) is diagonalizable. The main result of this paper gives necessary and sufficient conditions for a Schroeder solution F which has full rank near 0 without the added assumption of diagonalizability. More generally, it is proven in this paper that the functional equation F \circ \phi = \phi '(0)^k F with k a positive integer, is always solvable with an F whose component functions are linearly independent, but if k > 1 any such F cannot be injective near 0. In 2007 Enoch provides many theorems giving formal power series solutions to Schroeder's equation in several variables. It is also proved in this note that any formal power series solution indeed represents an analytic function on the ball.