A solution to Schroeder's equation in several variables
Abstract
Let &phis; be an analytic self-map of [special characters omitted], the unit ball in [special characters omitted], having 0 as the attracting fixed point, and having full-rank near 0. In the well-known case n = 1 (so [special characters omitted] is the disk) Schroeder's equation asks if there is an analytic f and a number λ satisfying f ∘ &phis; = λf. Under the current assumption that &phis;'(0) ≠ 0, Koenigs showed in 1884 that there is a solution f which is bijective near 0 precisely when λ = &phis;'(0). In 2003, Cowen and MacCluer formulated an analogous functional equation in several variables, i.e. for a non-negative integer n, by considering F ∘ &phis; = AF, for an n × n matrix A and [special characters omitted]-valued F. They showed that solving this equation with an F which has full rank near 0 is equivalent to solving F ∘ &phis; = &phis;'(0)F with such an F, and F ∘ &phis; = &phis;'(0)F shall be called Schroeder's equation in several variables. The main result of the 2003 Cowen and MacCluer paper provides necessary and sufficient conditions for an analytic solution, F, taking values in [special characters omitted] and having full-rank near 0, under the additional assumption that &phis;'(0) is diagonalizable. In 2007 Enoch provided many theorems giving formal power series solutions to Schroeder's equation in several variables. This thesis considers the more general equation, F ∘ &phis; = &phis;'(0) kF with k a positive integer, and proves there is always a solution F with linearly independent component functions, but that such an F cannot have full rank except possibly when k = 1. Furthermore when k = 1, and we are considering Schroeder's equation, necessary and sufficient conditions on &phis; are given to ensure F has has full rank near 0 without the added assumption of diagonalizability as needed in the 2003 Cowen and MacCluer paper. In response to Enoch's paper, it is proven in this thesis that any formal power series solution indeed represents an analytic function on the whole unit ball. How exactly resonance can lead to an obstruction of a full rank solution is discussed as well as some consequences of having solutions to Schroeder's equation.
Degree
Ph.D.
Advisors
Cowen, Purdue University.
Subject Area
Applied Mathematics|Mathematics
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