Formal power series solutions for Schroeder's equation in several complex variables
Abstract
In 1884, Koenigs showed that when ϕ is an analytic self-map of the unit disk fixing the origin, with 0 < :ϕ′(0): < 1, then Schroeder's functional equation, σ∘ϕ = ϕ ′(0)σ, can be solved for a unique analytic function σ in the disk with σ′(0) = 1. In 2003, C. Cowen and B. MacCluer considered an analogue of Schroeder's equation in the unit ball of [special characters omitted] for N > 1. Under some natural hypotheses, they gave necessary and sufficient conditions for the existence of an analytic solution σ satisfying σ′(0) = I when ϕ ′(0) is diagonalizable. In this dissertation, the problem when ϕ ′(0) is not diagonalizable is considered. Throughout this thesis, both ϕ(z) and σ(z) are vectors of purely formal power series. It will not be assumed that ϕ( z) is analytic or that the components of ϕ(z) or σ(z) converge. Nevertheless, because of a result in the aforementioned paper of C. Cowen and B. MacCluer, if the given ϕ( z) represents a map of the unit ball into itself of an appropriate form, then the results of this thesis can be used to produce solutions of Schroeder's equation that are composed of convergent power series, or possibly, to show that no such solutions exist. A method of matrix completion is used.
Degree
Ph.D.
Advisors
Cowen, Purdue University.
Subject Area
Mathematics
Off-Campus Purdue Users:
To access this dissertation, please log in to our
proxy server.